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Номер 13, страница 5 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Упражнения для повторения курса алгебры 7—9 классов - номер 13, страница 5.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113603 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => "6" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-13" "field_display_title" => "13" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ 0 => App\Models\Book {#1048 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:2 [ 0 => App\Models\Term {#1056 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1062 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1058} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1059} "field_country" => Illuminate\Database\Eloquent\Collection {#1061} "field_city" => Illuminate\Database\Eloquent\Collection {#1063} "field_series" => Illuminate\Database\Eloquent\Collection {#1065} "field_umk" => Illuminate\Database\Eloquent\Collection {#1066} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1067} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1069} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения для повторения курса алгебры 7—9 классов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "4" "field_branch_display" => "0" "field_branch_expanded" => "0" 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=> Illuminate\Database\Eloquent\Collection {#1042 #items: array:3 [ 0 => App\Models\Element {#1082 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1290663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1083 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Term {#1084 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113603" "type" => "task" ] "text" => "<p><strong>13.</strong> <strong>а)</strong> Первое натуральное число составляет 75% от второго натурального числа. Найдите натуральные числа, если значение их произведений равно 1200;</p><p><strong>б)</strong> если числитель дроби увеличить на 2, а знаменатель на 3, то получим дробь на $\frac{49}{35}$ больше данной дроби. Найдите первоначальную дробь;</p><p><strong>в)</strong> значение суммы цифр двузначного числа равно 9, а значение разности квадратов его цифр равно 27. Найдите двузначное число;</p><p><strong>г)</strong> катер проплыл по течению реки 36 км, против течения 48 км. На весь путь потрачено 6 ч времени. Найдите собственную скорость катера, если скорость течения реки равна 3 км/ч;</p><p><strong>д)</strong> промежуток 180 км первый поезд проходит на 1,5 ч быстрее, чем второй поезд. Найдите скорость каждого поезда, если поезда вместе за 3 ч проезжают 162 км.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "2806" "height" => 444 "path" => "/media/algebra_10/Abylkasymova-u/0-00/13-1.webp?ts=1753262978" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "2703" "height" => 1309 "path" => "/media/algebra_10/Abylkasymova-u/0-00/13-2.webp?ts=1753262978" ] ] ] #original: array:7 [ "id" => 1290663 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "task" => array:2 [ "refs" => "1113603" "type" => "task" ] "text" => "<p><strong>13.</strong> <strong>а)</strong> Первое натуральное число составляет 75% от второго натурального числа. Найдите натуральные числа, если значение их произведений равно 1200;</p><p><strong>б)</strong> если числитель дроби увеличить на 2, а знаменатель на 3, то получим дробь на $\frac{49}{35}$ больше данной дроби. Найдите первоначальную дробь;</p><p><strong>в)</strong> значение суммы цифр двузначного числа равно 9, а значение разности квадратов его цифр равно 27. Найдите двузначное число;</p><p><strong>г)</strong> катер проплыл по течению реки 36 км, против течения 48 км. На весь путь потрачено 6 ч времени. Найдите собственную скорость катера, если скорость течения реки равна 3 км/ч;</p><p><strong>д)</strong> промежуток 180 км первый поезд проходит на 1,5 ч быстрее, чем второй поезд. Найдите скорость каждого поезда, если поезда вместе за 3 ч проезжают 162 км.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "2806" "height" => 444 "path" => "/media/algebra_10/Abylkasymova-u/0-00/13-1.webp?ts=1753262978" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "2703" "height" => 1309 "path" => "/media/algebra_10/Abylkasymova-u/0-00/13-2.webp?ts=1753262978" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1291177 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Term {#1092 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1093} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1094} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1095} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1096} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113603" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1919" "height" => 2453 "path" => "/media/algebra_10/Abylkasymova-u/1-00/13-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "1876" "height" => 3535 "path" => "/media/algebra_10/Abylkasymova-u/1-00/13-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "13-3.jpg" "alt" => null "width" => "1927" "height" => 3249 "path" => "/media/algebra_10/Abylkasymova-u/1-00/13-3.webp?ts=1753263138" ] ] ] #original: array:6 [ "id" => 1291177 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1099} "task" => array:2 [ "refs" => "1113603" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1919" "height" => 2453 "path" => "/media/algebra_10/Abylkasymova-u/1-00/13-1.webp?ts=1753263138" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "1876" "height" => 3535 "path" => "/media/algebra_10/Abylkasymova-u/1-00/13-2.webp?ts=1753263138" ] 2 => array:5 [ "name" => "13-3.jpg" "alt" => null "width" => "1927" "height" => 3249 "path" => "/media/algebra_10/Abylkasymova-u/1-00/13-3.webp?ts=1753263138" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1097 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1549306 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1098 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Term {#1100 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1101} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1102} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1103} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113603" "type" => "task" ] "text" => "<p><strong>а)</strong> Пусть первое натуральное число равно $x$, а второе натуральное число равно $y$.</p><p>Из условия задачи известно, что первое число составляет 75% от второго, что можно записать в виде уравнения:$x = 0.75y$ или $x = \frac{3}{4}y$.</p><p>Также известно, что произведение этих чисел равно 1200:$x \cdot y = 1200$.</p><p>Подставим первое уравнение во второе:$(\frac{3}{4}y) \cdot y = 1200$</p><p>$\frac{3}{4}y^2 = 1200$</p><p>$y^2 = \frac{1200 \cdot 4}{3}$</p><p>$y^2 = 400 \cdot 4$</p><p>$y^2 = 1600$</p><p>Так как $y$ — натуральное число, $y = \sqrt{1600} = 40$.</p><p>Теперь найдем $x$, подставив значение $y$ в первое уравнение:$x = \frac{3}{4} \cdot 40 = 3 \cdot 10 = 30$.</p><p>Искомые натуральные числа — 30 и 40.Проверка: $30 \cdot 40 = 1200$. $30 / 40 = 0.75 = 75\%$.</p><p><strong>Ответ:</strong> 30 и 40.</p><p><strong>б)</strong> Пусть первоначальная дробь равна $\frac{x}{y}$.</p><p>Если числитель увеличить на 2, а знаменатель на 3, получим новую дробь $\frac{x+2}{y+3}$.</p><p>В условии сказано, что новая дробь на $\frac{49}{35}$ больше данной. Запишем уравнение:$\frac{x+2}{y+3} = \frac{x}{y} + \frac{49}{35}$.</p><p>Упростим дробь $\frac{49}{35} = \frac{7}{5}$.$\frac{x+2}{y+3} = \frac{x}{y} + \frac{7}{5}$.</p><p>Преобразуем уравнение:$\frac{x+2}{y+3} - \frac{x}{y} = \frac{7}{5}$</p><p>$\frac{y(x+2) - x(y+3)}{y(y+3)} = \frac{7}{5}$</p><p>$\frac{xy+2y-xy-3x}{y^2+3y} = \frac{7}{5}$</p><p>$\frac{2y-3x}{y^2+3y} = \frac{7}{5}$</p><p>$5(2y-3x) = 7(y^2+3y)$</p><p>$10y - 15x = 7y^2 + 21y$</p><p>$7y^2 + 11y + 15x = 0$</p><p>Для натуральных (положительных) чисел $x$ и $y$ сумма $7y^2 + 11y + 15x$ всегда будет положительной и не может равняться нулю. Это означает, что в условии задачи, скорее всего, опечатка. Наиболее вероятная опечатка — в значении $\frac{49}{35}$. Предположим, что имелось в виду $\frac{4}{35}$.</p><p>Тогда уравнение примет вид:$\frac{2y-3x}{y^2+3y} = \frac{4}{35}$</p><p>$35(2y-3x) = 4(y^2+3y)$</p><p>$70y - 105x = 4y^2 + 12y$</p><p>$105x = 58y - 4y^2$</p><p>$105x = 2y(29-2y)$</p><p>Поскольку $x$ и $y$ должны быть натуральными числами, $x > 0$, следовательно $29-2y > 0$, что означает $2y < 29$ или $y \le 14$.</p><p>Правая часть $2y(29-2y)$ должна быть делимой на 105, то есть на 3, 5 и 7.Проверим возможные значения $y$ от 1 до 14.Если $y=7$:$105x = 2 \cdot 7 \cdot (29 - 2 \cdot 7) = 14 \cdot (29 - 14) = 14 \cdot 15 = 210$.</p><p>$x = \frac{210}{105} = 2$.</p><p>Получили натуральные $x=2$ и $y=7$. Первоначальная дробь — $\frac{2}{7}$.</p><p>Проверка: новая дробь $\frac{2+2}{7+3} = \frac{4}{10} = \frac{2}{5}$. Разность $\frac{2}{5} - \frac{2}{7} = \frac{14-10}{35} = \frac{4}{35}$. Условие выполнено.</p><p><strong>Ответ:</strong> $\frac{2}{7}$.</p><p><strong>в)</strong> Пусть двузначное число состоит из цифры десятков $a$ и цифры единиц $b$. Значение числа равно $10a+b$.Из условия задачи имеем систему уравнений:1. Сумма цифр равна 9: $a+b=9$.2. Разность квадратов цифр равна 27: $a^2-b^2=27$. (Так как разность положительна, квадрат цифры десятков больше квадрата цифры единиц, $a > b$).</p><p>Разложим второе уравнение по формуле разности квадратов:$(a-b)(a+b)=27$.</p><p>Подставим в это уравнение значение $a+b$ из первого уравнения:$(a-b) \cdot 9 = 27$.</p><p>$a-b = \frac{27}{9} = 3$.</p><p>Теперь у нас есть простая система из двух линейных уравнений:$\begin{cases} a+b=9 \\ a-b=3 \end{cases}$</p><p>Сложим два уравнения:$(a+b)+(a-b) = 9+3$</p><p>$2a = 12$</p><p>$a = 6$.</p><p>Подставим значение $a$ в первое уравнение:$6+b=9$</p><p>$b=3$.</p><p>Цифра десятков равна 6, цифра единиц равна 3. Искомое число — 63.</p><p><strong>Ответ:</strong> 63.</p><p><strong>г)</strong> Пусть $v_c$ — собственная скорость катера в км/ч. Скорость течения реки $v_p = 3$ км/ч.</p><p>Скорость катера по течению: $v_{по} = v_c + v_p = v_c + 3$ км/ч.</p><p>Скорость катера против течения: $v_{против} = v_c - v_p = v_c - 3$ км/ч. (При этом $v_c > 3$).</p><p>Время движения по течению: $t_{по} = \frac{S_{по}}{v_{по}} = \frac{36}{v_c+3}$ ч.</p><p>Время движения против течения: $t_{против} = \frac{S_{против}}{v_{против}} = \frac{48}{v_c-3}$ ч.</p><p>Общее время в пути составляет 6 часов:$t_{по} + t_{против} = 6$</p><p>$\frac{36}{v_c+3} + \frac{48}{v_c-3} = 6$.</p><p>Разделим все уравнение на 6 для упрощения:$\frac{6}{v_c+3} + \frac{8}{v_c-3} = 1$.</p><p>Приведем к общему знаменателю $(v_c+3)(v_c-3)$:$6(v_c-3) + 8(v_c+3) = (v_c+3)(v_c-3)$</p><p>$6v_c - 18 + 8v_c + 24 = v_c^2 - 9$</p><p>$14v_c + 6 = v_c^2 - 9$</p><p>$v_c^2 - 14v_c - 15 = 0$.</p><p>Решим квадратное уравнение. Дискриминант $D = b^2 - 4ac = (-14)^2 - 4 \cdot 1 \cdot (-15) = 196 + 60 = 256$.</p><p>$v_c = \frac{-b \pm \sqrt{D}}{2a} = \frac{14 \pm \sqrt{256}}{2} = \frac{14 \pm 16}{2}$.</p><p>Получаем два корня:$v_{c1} = \frac{14+16}{2} = \frac{30}{2} = 15$.$v_{c2} = \frac{14-16}{2} = \frac{-2}{2} = -1$.</p><p>Скорость не может быть отрицательной, поэтому $v_c = 15$ км/ч. Это значение удовлетворяет условию $v_c > 3$.</p><p><strong>Ответ:</strong> 15 км/ч.</p><p><strong>д)</strong> Пусть $v_1$ — скорость первого поезда (в км/ч), а $v_2$ — скорость второго поезда (в км/ч).</p><p>Из первого условия: время первого поезда для прохождения 180 км ($t_1 = \frac{180}{v_1}$) на 1,5 часа меньше времени второго ($t_2 = \frac{180}{v_2}$):$\frac{180}{v_2} - \frac{180}{v_1} = 1.5$.</p><p>Из второго условия: вместе за 3 часа поезда проезжают 162 км. Это означает, что сумма их скоростей, умноженная на 3, равна 162:$3(v_1 + v_2) = 162$.</p><p>Из второго уравнения находим сумму скоростей:$v_1 + v_2 = \frac{162}{3} = 54$.</p><p>Отсюда $v_1 = 54 - v_2$.</p><p>Подставим это выражение в первое уравнение:$\frac{180}{v_2} - \frac{180}{54 - v_2} = 1.5$.</p><p>Разделим уравнение на 1.5 (или умножим на 2/3):$\frac{120}{v_2} - \frac{120}{54 - v_2} = 1$.</p><p>Приведем к общему знаменателю $v_2(54-v_2)$:$120(54 - v_2) - 120v_2 = v_2(54 - v_2)$</p><p>$6480 - 120v_2 - 120v_2 = 54v_2 - v_2^2$</p><p>$6480 - 240v_2 = 54v_2 - v_2^2$</p><p>$v_2^2 - 294v_2 + 6480 = 0$.</p><p>Решим квадратное уравнение. Дискриминант $D = b^2 - 4ac = (-294)^2 - 4 \cdot 1 \cdot 6480 = 86436 - 25920 = 60516$.</p><p>$\sqrt{D} = \sqrt{60516} = 246$.</p><p>$v_2 = \frac{294 \pm 246}{2}$.</p><p>Получаем два корня для $v_2$:$v_{2_a} = \frac{294+246}{2} = \frac{540}{2} = 270$.$v_{2_b} = \frac{294-246}{2} = \frac{48}{2} = 24$.</p><p>Рассмотрим оба случая:1. Если $v_2 = 270$ км/ч, то $v_1 = 54 - 270 = -216$ км/ч. Скорость не может быть отрицательной, этот корень не подходит.2. Если $v_2 = 24$ км/ч, то $v_1 = 54 - 24 = 30$ км/ч. Оба значения положительны и являются решением.</p><p>Проверка: $t_1 = 180/30 = 6$ ч. $t_2 = 180/24 = 7.5$ ч. $t_2 - t_1 = 7.5 - 6 = 1.5$ ч. Условие выполняется.</p><p><strong>Ответ:</strong> скорость первого поезда 30 км/ч, скорость второго поезда 24 км/ч.</p>" ] #original: array:6 [ "id" => 1549306 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1113603" "type" => "task" ] "text" => "<p><strong>а)</strong> Пусть первое натуральное число равно $x$, а второе натуральное число равно $y$.</p><p>Из условия задачи известно, что первое число составляет 75% от второго, что можно записать в виде уравнения:$x = 0.75y$ или $x = \frac{3}{4}y$.</p><p>Также известно, что произведение этих чисел равно 1200:$x \cdot y = 1200$.</p><p>Подставим первое уравнение во второе:$(\frac{3}{4}y) \cdot y = 1200$</p><p>$\frac{3}{4}y^2 = 1200$</p><p>$y^2 = \frac{1200 \cdot 4}{3}$</p><p>$y^2 = 400 \cdot 4$</p><p>$y^2 = 1600$</p><p>Так как $y$ — натуральное число, $y = \sqrt{1600} = 40$.</p><p>Теперь найдем $x$, подставив значение $y$ в первое уравнение:$x = \frac{3}{4} \cdot 40 = 3 \cdot 10 = 30$.</p><p>Искомые натуральные числа — 30 и 40.Проверка: $30 \cdot 40 = 1200$. $30 / 40 = 0.75 = 75\%$.</p><p><strong>Ответ:</strong> 30 и 40.</p><p><strong>б)</strong> Пусть первоначальная дробь равна $\frac{x}{y}$.</p><p>Если числитель увеличить на 2, а знаменатель на 3, получим новую дробь $\frac{x+2}{y+3}$.</p><p>В условии сказано, что новая дробь на $\frac{49}{35}$ больше данной. Запишем уравнение:$\frac{x+2}{y+3} = \frac{x}{y} + \frac{49}{35}$.</p><p>Упростим дробь $\frac{49}{35} = \frac{7}{5}$.$\frac{x+2}{y+3} = \frac{x}{y} + \frac{7}{5}$.</p><p>Преобразуем уравнение:$\frac{x+2}{y+3} - \frac{x}{y} = \frac{7}{5}$</p><p>$\frac{y(x+2) - x(y+3)}{y(y+3)} = \frac{7}{5}$</p><p>$\frac{xy+2y-xy-3x}{y^2+3y} = \frac{7}{5}$</p><p>$\frac{2y-3x}{y^2+3y} = \frac{7}{5}$</p><p>$5(2y-3x) = 7(y^2+3y)$</p><p>$10y - 15x = 7y^2 + 21y$</p><p>$7y^2 + 11y + 15x = 0$</p><p>Для натуральных (положительных) чисел $x$ и $y$ сумма $7y^2 + 11y + 15x$ всегда будет положительной и не может равняться нулю. Это означает, что в условии задачи, скорее всего, опечатка. Наиболее вероятная опечатка — в значении $\frac{49}{35}$. Предположим, что имелось в виду $\frac{4}{35}$.</p><p>Тогда уравнение примет вид:$\frac{2y-3x}{y^2+3y} = \frac{4}{35}$</p><p>$35(2y-3x) = 4(y^2+3y)$</p><p>$70y - 105x = 4y^2 + 12y$</p><p>$105x = 58y - 4y^2$</p><p>$105x = 2y(29-2y)$</p><p>Поскольку $x$ и $y$ должны быть натуральными числами, $x > 0$, следовательно $29-2y > 0$, что означает $2y < 29$ или $y \le 14$.</p><p>Правая часть $2y(29-2y)$ должна быть делимой на 105, то есть на 3, 5 и 7.Проверим возможные значения $y$ от 1 до 14.Если $y=7$:$105x = 2 \cdot 7 \cdot (29 - 2 \cdot 7) = 14 \cdot (29 - 14) = 14 \cdot 15 = 210$.</p><p>$x = \frac{210}{105} = 2$.</p><p>Получили натуральные $x=2$ и $y=7$. Первоначальная дробь — $\frac{2}{7}$.</p><p>Проверка: новая дробь $\frac{2+2}{7+3} = \frac{4}{10} = \frac{2}{5}$. Разность $\frac{2}{5} - \frac{2}{7} = \frac{14-10}{35} = \frac{4}{35}$. Условие выполнено.</p><p><strong>Ответ:</strong> $\frac{2}{7}$.</p><p><strong>в)</strong> Пусть двузначное число состоит из цифры десятков $a$ и цифры единиц $b$. Значение числа равно $10a+b$.Из условия задачи имеем систему уравнений:1. Сумма цифр равна 9: $a+b=9$.2. Разность квадратов цифр равна 27: $a^2-b^2=27$. (Так как разность положительна, квадрат цифры десятков больше квадрата цифры единиц, $a > b$).</p><p>Разложим второе уравнение по формуле разности квадратов:$(a-b)(a+b)=27$.</p><p>Подставим в это уравнение значение $a+b$ из первого уравнения:$(a-b) \cdot 9 = 27$.</p><p>$a-b = \frac{27}{9} = 3$.</p><p>Теперь у нас есть простая система из двух линейных уравнений:$\begin{cases} a+b=9 \\ a-b=3 \end{cases}$</p><p>Сложим два уравнения:$(a+b)+(a-b) = 9+3$</p><p>$2a = 12$</p><p>$a = 6$.</p><p>Подставим значение $a$ в первое уравнение:$6+b=9$</p><p>$b=3$.</p><p>Цифра десятков равна 6, цифра единиц равна 3. Искомое число — 63.</p><p><strong>Ответ:</strong> 63.</p><p><strong>г)</strong> Пусть $v_c$ — собственная скорость катера в км/ч. Скорость течения реки $v_p = 3$ км/ч.</p><p>Скорость катера по течению: $v_{по} = v_c + v_p = v_c + 3$ км/ч.</p><p>Скорость катера против течения: $v_{против} = v_c - v_p = v_c - 3$ км/ч. (При этом $v_c > 3$).</p><p>Время движения по течению: $t_{по} = \frac{S_{по}}{v_{по}} = \frac{36}{v_c+3}$ ч.</p><p>Время движения против течения: $t_{против} = \frac{S_{против}}{v_{против}} = \frac{48}{v_c-3}$ ч.</p><p>Общее время в пути составляет 6 часов:$t_{по} + t_{против} = 6$</p><p>$\frac{36}{v_c+3} + \frac{48}{v_c-3} = 6$.</p><p>Разделим все уравнение на 6 для упрощения:$\frac{6}{v_c+3} + \frac{8}{v_c-3} = 1$.</p><p>Приведем к общему знаменателю $(v_c+3)(v_c-3)$:$6(v_c-3) + 8(v_c+3) = (v_c+3)(v_c-3)$</p><p>$6v_c - 18 + 8v_c + 24 = v_c^2 - 9$</p><p>$14v_c + 6 = v_c^2 - 9$</p><p>$v_c^2 - 14v_c - 15 = 0$.</p><p>Решим квадратное уравнение. Дискриминант $D = b^2 - 4ac = (-14)^2 - 4 \cdot 1 \cdot (-15) = 196 + 60 = 256$.</p><p>$v_c = \frac{-b \pm \sqrt{D}}{2a} = \frac{14 \pm \sqrt{256}}{2} = \frac{14 \pm 16}{2}$.</p><p>Получаем два корня:$v_{c1} = \frac{14+16}{2} = \frac{30}{2} = 15$.$v_{c2} = \frac{14-16}{2} = \frac{-2}{2} = -1$.</p><p>Скорость не может быть отрицательной, поэтому $v_c = 15$ км/ч. Это значение удовлетворяет условию $v_c > 3$.</p><p><strong>Ответ:</strong> 15 км/ч.</p><p><strong>д)</strong> Пусть $v_1$ — скорость первого поезда (в км/ч), а $v_2$ — скорость второго поезда (в км/ч).</p><p>Из первого условия: время первого поезда для прохождения 180 км ($t_1 = \frac{180}{v_1}$) на 1,5 часа меньше времени второго ($t_2 = \frac{180}{v_2}$):$\frac{180}{v_2} - \frac{180}{v_1} = 1.5$.</p><p>Из второго условия: вместе за 3 часа поезда проезжают 162 км. Это означает, что сумма их скоростей, умноженная на 3, равна 162:$3(v_1 + v_2) = 162$.</p><p>Из второго уравнения находим сумму скоростей:$v_1 + v_2 = \frac{162}{3} = 54$.</p><p>Отсюда $v_1 = 54 - v_2$.</p><p>Подставим это выражение в первое уравнение:$\frac{180}{v_2} - \frac{180}{54 - v_2} = 1.5$.</p><p>Разделим уравнение на 1.5 (или умножим на 2/3):$\frac{120}{v_2} - \frac{120}{54 - v_2} = 1$.</p><p>Приведем к общему знаменателю $v_2(54-v_2)$:$120(54 - v_2) - 120v_2 = v_2(54 - v_2)$</p><p>$6480 - 120v_2 - 120v_2 = 54v_2 - v_2^2$</p><p>$6480 - 240v_2 = 54v_2 - v_2^2$</p><p>$v_2^2 - 294v_2 + 6480 = 0$.</p><p>Решим квадратное уравнение. Дискриминант $D = b^2 - 4ac = (-294)^2 - 4 \cdot 1 \cdot 6480 = 86436 - 25920 = 60516$.</p><p>$\sqrt{D} = \sqrt{60516} = 246$.</p><p>$v_2 = \frac{294 \pm 246}{2}$.</p><p>Получаем два корня для $v_2$:$v_{2_a} = \frac{294+246}{2} = \frac{540}{2} = 270$.$v_{2_b} = \frac{294-246}{2} = \frac{48}{2} = 24$.</p><p>Рассмотрим оба случая:1. Если $v_2 = 270$ км/ч, то $v_1 = 54 - 270 = -216$ км/ч. Скорость не может быть отрицательной, этот корень не подходит.2. Если $v_2 = 24$ км/ч, то $v_1 = 54 - 24 = 30$ км/ч. Оба значения положительны и являются решением.</p><p>Проверка: $t_1 = 180/30 = 6$ ч. $t_2 = 180/24 = 7.5$ ч. $t_2 - t_1 = 7.5 - 6 = 1.5$ ч. Условие выполняется.</p><p><strong>Ответ:</strong> скорость первого поезда 30 км/ч, скорость второго поезда 24 км/ч.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113604" "type" => "task" ] "previous" => array:2 [ "refs" => "1113602" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\BookPage {#1040 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1098342 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-5" "field_display_title" => "5" "field_folder" => "1" "field_image_name" => "5" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Element {#1116 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Edition {#1083} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "3154" "height" => 4601 "path" => "/media/algebra_10/Abylkasymova-u/0-1/5-1.webp?ts=1752147445" ] ] ] #original: array:6 [ "id" => 1261767 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1118} "book_page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "5-1.jpg" "alt" => null "width" => "3154" "height" => 4601 "path" => "/media/algebra_10/Abylkasymova-u/0-1/5-1.webp?ts=1752147445" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1098343" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098341" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:8 [ 0 => App\Models\Task {#1191 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1193 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:3 [ 0 => App\Models\Element {#1207 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1209 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] …24 } 2 => App\Models\Element {#1211 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113597" "type" => "task" ] "previous" => array:2 [ "refs" => "1113595" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1204 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113596 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1192} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1193} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1194} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1195} "content" => Illuminate\Database\Eloquent\Collection {#1198} "next" => array:2 [ "refs" => "1113597" "type" => "task" ] "previous" => array:2 [ "refs" => "1113595" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1204} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1197 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113597 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1203 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:3 [ 0 => App\Models\Element {#1221 …30} 1 => App\Models\Element {#1223 …30} 2 => App\Models\Element {#1225 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113598" "type" => "task" ] "previous" => array:2 [ "refs" => "1113596" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113597 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-7" "field_display_title" => "7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1196} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1203} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1205} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1201} "content" => Illuminate\Database\Eloquent\Collection {#1200} "next" => array:2 [ "refs" => "1113598" "type" => "task" ] "previous" => array:2 [ "refs" => "1113596" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1218} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1202 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1217 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:3 [ 0 => App\Models\Element {#1235 …30} 1 => App\Models\Element {#1237 …30} 2 => App\Models\Element {#1239 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113599" "type" => "task" ] "previous" => array:2 [ "refs" => "1113597" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113598 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1199} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1217} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1219} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215} "content" => Illuminate\Database\Eloquent\Collection {#1214} "next" => array:2 [ "refs" => "1113599" "type" => "task" ] "previous" => array:2 [ "refs" => "1113597" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1232} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1216 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113599 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1231 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:3 [ 0 => App\Models\Element {#1249 …30} 1 => App\Models\Element {#1251 …30} 2 => App\Models\Element {#1253 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113600" "type" => "task" ] "previous" => array:2 [ "refs" => "1113598" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1246 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113599 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1213} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1231} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1233} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1229} "content" => Illuminate\Database\Eloquent\Collection {#1228} "next" => array:2 [ "refs" => "1113600" "type" => "task" ] "previous" => array:2 [ "refs" => "1113598" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1246} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1230 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1245 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:3 [ 0 => App\Models\Element {#1263 …30} 1 => App\Models\Element {#1265 …30} 2 => App\Models\Element {#1267 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113601" "type" => "task" ] "previous" => array:2 [ "refs" => "1113599" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113600 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-10" "field_display_title" => "10" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1245} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1247} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243} "content" => Illuminate\Database\Eloquent\Collection {#1242} "next" => array:2 [ "refs" => "1113601" "type" => "task" ] "previous" => array:2 [ "refs" => "1113599" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1260} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1244 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113601 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1259 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1261 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1257 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:3 [ 0 => App\Models\Element {#1277 …30} 1 => App\Models\Element {#1279 …30} 2 => App\Models\Element {#1281 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113602" "type" => "task" ] "previous" => array:2 [ "refs" => "1113600" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1274 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113601 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/00-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1241} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1259} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1261} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1257} "content" => Illuminate\Database\Eloquent\Collection {#1256} "next" => array:2 [ "refs" => "1113602" "type" => "task" ] "previous" => array:2 [ "refs" => "1113600" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1274} "page" => array:2 [ "refs" => "1098342" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1258 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113602 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "5" "field_page_end" => null "field_url" => 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№13 (с. 5)
Условие. №13 (с. 5)
скриншот условия
13. а) Первое натуральное число составляет 75% от второго натурального числа. Найдите натуральные числа, если значение их произведений равно 1200;
б) если числитель дроби увеличить на 2, а знаменатель на 3, то получим дробь на $\frac{49}{35}$ больше данной дроби. Найдите первоначальную дробь;
в) значение суммы цифр двузначного числа равно 9, а значение разности квадратов его цифр равно 27. Найдите двузначное число;
г) катер проплыл по течению реки 36 км, против течения 48 км. На весь путь потрачено 6 ч времени. Найдите собственную скорость катера, если скорость течения реки равна 3 км/ч;
д) промежуток 180 км первый поезд проходит на 1,5 ч быстрее, чем второй поезд. Найдите скорость каждого поезда, если поезда вместе за 3 ч проезжают 162 км.
Решение. №13 (с. 5)
Решение 2. №13 (с. 5)
а) Пусть первое натуральное число равно $x$, а второе натуральное число равно $y$.
Из условия задачи известно, что первое число составляет 75% от второго, что можно записать в виде уравнения:$x = 0.75y$ или $x = \frac{3}{4}y$.
Также известно, что произведение этих чисел равно 1200:$x \cdot y = 1200$.
Подставим первое уравнение во второе:$(\frac{3}{4}y) \cdot y = 1200$
$\frac{3}{4}y^2 = 1200$
$y^2 = \frac{1200 \cdot 4}{3}$
$y^2 = 400 \cdot 4$
$y^2 = 1600$
Так как $y$ — натуральное число, $y = \sqrt{1600} = 40$.
Теперь найдем $x$, подставив значение $y$ в первое уравнение:$x = \frac{3}{4} \cdot 40 = 3 \cdot 10 = 30$.
Искомые натуральные числа — 30 и 40.Проверка: $30 \cdot 40 = 1200$. $30 / 40 = 0.75 = 75\%$.
Ответ: 30 и 40.
б) Пусть первоначальная дробь равна $\frac{x}{y}$.
Если числитель увеличить на 2, а знаменатель на 3, получим новую дробь $\frac{x+2}{y+3}$.
В условии сказано, что новая дробь на $\frac{49}{35}$ больше данной. Запишем уравнение:$\frac{x+2}{y+3} = \frac{x}{y} + \frac{49}{35}$.
Упростим дробь $\frac{49}{35} = \frac{7}{5}$.$\frac{x+2}{y+3} = \frac{x}{y} + \frac{7}{5}$.
Преобразуем уравнение:$\frac{x+2}{y+3} - \frac{x}{y} = \frac{7}{5}$
$\frac{y(x+2) - x(y+3)}{y(y+3)} = \frac{7}{5}$
$\frac{xy+2y-xy-3x}{y^2+3y} = \frac{7}{5}$
$\frac{2y-3x}{y^2+3y} = \frac{7}{5}$
$5(2y-3x) = 7(y^2+3y)$
$10y - 15x = 7y^2 + 21y$
$7y^2 + 11y + 15x = 0$
Для натуральных (положительных) чисел $x$ и $y$ сумма $7y^2 + 11y + 15x$ всегда будет положительной и не может равняться нулю. Это означает, что в условии задачи, скорее всего, опечатка. Наиболее вероятная опечатка — в значении $\frac{49}{35}$. Предположим, что имелось в виду $\frac{4}{35}$.
Тогда уравнение примет вид:$\frac{2y-3x}{y^2+3y} = \frac{4}{35}$
$35(2y-3x) = 4(y^2+3y)$
$70y - 105x = 4y^2 + 12y$
$105x = 58y - 4y^2$
$105x = 2y(29-2y)$
Поскольку $x$ и $y$ должны быть натуральными числами, $x > 0$, следовательно $29-2y > 0$, что означает $2y < 29$ или $y \le 14$.
Правая часть $2y(29-2y)$ должна быть делимой на 105, то есть на 3, 5 и 7.Проверим возможные значения $y$ от 1 до 14.Если $y=7$:$105x = 2 \cdot 7 \cdot (29 - 2 \cdot 7) = 14 \cdot (29 - 14) = 14 \cdot 15 = 210$.
$x = \frac{210}{105} = 2$.
Получили натуральные $x=2$ и $y=7$. Первоначальная дробь — $\frac{2}{7}$.
Проверка: новая дробь $\frac{2+2}{7+3} = \frac{4}{10} = \frac{2}{5}$. Разность $\frac{2}{5} - \frac{2}{7} = \frac{14-10}{35} = \frac{4}{35}$. Условие выполнено.
Ответ: $\frac{2}{7}$.
в) Пусть двузначное число состоит из цифры десятков $a$ и цифры единиц $b$. Значение числа равно $10a+b$.Из условия задачи имеем систему уравнений:1. Сумма цифр равна 9: $a+b=9$.2. Разность квадратов цифр равна 27: $a^2-b^2=27$. (Так как разность положительна, квадрат цифры десятков больше квадрата цифры единиц, $a > b$).
Разложим второе уравнение по формуле разности квадратов:$(a-b)(a+b)=27$.
Подставим в это уравнение значение $a+b$ из первого уравнения:$(a-b) \cdot 9 = 27$.
$a-b = \frac{27}{9} = 3$.
Теперь у нас есть простая система из двух линейных уравнений:$\begin{cases} a+b=9 \\ a-b=3 \end{cases}$
Сложим два уравнения:$(a+b)+(a-b) = 9+3$
$2a = 12$
$a = 6$.
Подставим значение $a$ в первое уравнение:$6+b=9$
$b=3$.
Цифра десятков равна 6, цифра единиц равна 3. Искомое число — 63.
Ответ: 63.
г) Пусть $v_c$ — собственная скорость катера в км/ч. Скорость течения реки $v_p = 3$ км/ч.
Скорость катера по течению: $v_{по} = v_c + v_p = v_c + 3$ км/ч.
Скорость катера против течения: $v_{против} = v_c - v_p = v_c - 3$ км/ч. (При этом $v_c > 3$).
Время движения по течению: $t_{по} = \frac{S_{по}}{v_{по}} = \frac{36}{v_c+3}$ ч.
Время движения против течения: $t_{против} = \frac{S_{против}}{v_{против}} = \frac{48}{v_c-3}$ ч.
Общее время в пути составляет 6 часов:$t_{по} + t_{против} = 6$
$\frac{36}{v_c+3} + \frac{48}{v_c-3} = 6$.
Разделим все уравнение на 6 для упрощения:$\frac{6}{v_c+3} + \frac{8}{v_c-3} = 1$.
Приведем к общему знаменателю $(v_c+3)(v_c-3)$:$6(v_c-3) + 8(v_c+3) = (v_c+3)(v_c-3)$
$6v_c - 18 + 8v_c + 24 = v_c^2 - 9$
$14v_c + 6 = v_c^2 - 9$
$v_c^2 - 14v_c - 15 = 0$.
Решим квадратное уравнение. Дискриминант $D = b^2 - 4ac = (-14)^2 - 4 \cdot 1 \cdot (-15) = 196 + 60 = 256$.
$v_c = \frac{-b \pm \sqrt{D}}{2a} = \frac{14 \pm \sqrt{256}}{2} = \frac{14 \pm 16}{2}$.
Получаем два корня:$v_{c1} = \frac{14+16}{2} = \frac{30}{2} = 15$.$v_{c2} = \frac{14-16}{2} = \frac{-2}{2} = -1$.
Скорость не может быть отрицательной, поэтому $v_c = 15$ км/ч. Это значение удовлетворяет условию $v_c > 3$.
Ответ: 15 км/ч.
д) Пусть $v_1$ — скорость первого поезда (в км/ч), а $v_2$ — скорость второго поезда (в км/ч).
Из первого условия: время первого поезда для прохождения 180 км ($t_1 = \frac{180}{v_1}$) на 1,5 часа меньше времени второго ($t_2 = \frac{180}{v_2}$):$\frac{180}{v_2} - \frac{180}{v_1} = 1.5$.
Из второго условия: вместе за 3 часа поезда проезжают 162 км. Это означает, что сумма их скоростей, умноженная на 3, равна 162:$3(v_1 + v_2) = 162$.
Из второго уравнения находим сумму скоростей:$v_1 + v_2 = \frac{162}{3} = 54$.
Отсюда $v_1 = 54 - v_2$.
Подставим это выражение в первое уравнение:$\frac{180}{v_2} - \frac{180}{54 - v_2} = 1.5$.
Разделим уравнение на 1.5 (или умножим на 2/3):$\frac{120}{v_2} - \frac{120}{54 - v_2} = 1$.
Приведем к общему знаменателю $v_2(54-v_2)$:$120(54 - v_2) - 120v_2 = v_2(54 - v_2)$
$6480 - 120v_2 - 120v_2 = 54v_2 - v_2^2$
$6480 - 240v_2 = 54v_2 - v_2^2$
$v_2^2 - 294v_2 + 6480 = 0$.
Решим квадратное уравнение. Дискриминант $D = b^2 - 4ac = (-294)^2 - 4 \cdot 1 \cdot 6480 = 86436 - 25920 = 60516$.
$\sqrt{D} = \sqrt{60516} = 246$.
$v_2 = \frac{294 \pm 246}{2}$.
Получаем два корня для $v_2$:$v_{2_a} = \frac{294+246}{2} = \frac{540}{2} = 270$.$v_{2_b} = \frac{294-246}{2} = \frac{48}{2} = 24$.
Рассмотрим оба случая:1. Если $v_2 = 270$ км/ч, то $v_1 = 54 - 270 = -216$ км/ч. Скорость не может быть отрицательной, этот корень не подходит.2. Если $v_2 = 24$ км/ч, то $v_1 = 54 - 24 = 30$ км/ч. Оба значения положительны и являются решением.
Проверка: $t_1 = 180/30 = 6$ ч. $t_2 = 180/24 = 7.5$ ч. $t_2 - t_1 = 7.5 - 6 = 1.5$ ч. Условие выполняется.
Ответ: скорость первого поезда 30 км/ч, скорость второго поезда 24 км/ч.
Другие задания:
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