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Номер 111, страница 38 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
I. Уравнения, неравенства с двумя переменными и их системы. 3. Решение текстовых задач с использованием систем нелинейных уравнений с двумя переменными - номер 111, страница 38.
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" => 1107808 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-111" "field_display_title" => "111" "field_outside_task" => null "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" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "18" "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 {#1078 #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" => 4298 "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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #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" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #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:3 [ 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" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "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" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "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 => "*" ] } 2 => App\Models\Term {#1058 #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" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "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 {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #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 учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #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" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "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 {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "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 учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #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 {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "18" "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 {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #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" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "18" "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 {#1115 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114} "field_page_start" => "18" "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 {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #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\Branch {#1117 #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" => 1107642 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3. Решение текстовых задач с использованием систем нелинейных уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "34" "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 {#1119 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107642 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3. Решение текстовых задач с использованием систем нелинейных уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "34" "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 {#1119} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #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" => 1276466 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1087 #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" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1043 #items: 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 {#1041 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1043} "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 {#1041} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081} "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" => "1107808" "type" => "task" ] "text" => "<p><strong>111.</strong> Найдите:<br><strong>а)</strong> сторону ромба, площадь которого равна $80 \text{ см}^2$, а отношение диагоналей равно 0,8;<br><strong>б)</strong> площадь ромба, периметр которого равен 2 дм, а диагонали относятся как 3 : 4.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "111-1.jpg" "alt" => null "width" => "1569" "height" => 276 "path" => "/media/algebra_09/soltan-u19/0-1/111-1.webp?ts=1752834175" ] ] ] #original: array:7 [ "id" => 1276466 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "1107808" "type" => "task" ] "text" => "<p><strong>111.</strong> Найдите:<br><strong>а)</strong> сторону ромба, площадь которого равна $80 \text{ см}^2$, а отношение диагоналей равно 0,8;<br><strong>б)</strong> площадь ромба, периметр которого равен 2 дм, а диагонали относятся как 3 : 4.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "111-1.jpg" "alt" => null "width" => "1569" "height" => 276 "path" => "/media/algebra_09/soltan-u19/0-1/111-1.webp?ts=1752834175" ] ] ] #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 {#1093 #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" => 1277728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 #items: 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 {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088} "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 {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091} "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" => "1107808" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "111-1.jpg" "alt" => null "width" => "1304" "height" => 2429 "path" => "/media/algebra_09/soltan-u19/1-1/111-1.webp?ts=1752830836" ] 1 => array:5 [ "name" => "111-2.jpg" "alt" => null "width" => "904" "height" => 667 "path" => "/media/algebra_09/soltan-u19/1-1/111-2.webp?ts=1752830836" ] ] ] #original: array:6 [ "id" => 1277728 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1107808" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "111-1.jpg" "alt" => null "width" => "1304" "height" => 2429 "path" => "/media/algebra_09/soltan-u19/1-1/111-1.webp?ts=1752830836" ] 1 => array:5 [ "name" => "111-2.jpg" "alt" => null "width" => "904" "height" => 667 "path" => "/media/algebra_09/soltan-u19/1-1/111-2.webp?ts=1752830836" ] ] ] #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 {#1084 #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" => 1553712 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1080 #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" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 #items: 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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034} "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 {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108} "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" => "1107808" "type" => "task" ] "text" => "<p><strong>а)</strong>Площадь ромба $S$ вычисляется через его диагонали $d_1$ и $d_2$ по формуле $S = \frac{d_1 \cdot d_2}{2}$. По условию задачи, площадь ромба равна 80 см², а отношение диагоналей $\frac{d_1}{d_2} = 0,8$.Из соотношения диагоналей выразим одну через другую: $d_1 = 0,8 \cdot d_2$.Подставим это выражение в формулу площади:$80 = \frac{(0,8 \cdot d_2) \cdot d_2}{2}$$160 = 0,8 \cdot d_2^2$$d_2^2 = \frac{160}{0,8} = \frac{1600}{8} = 200$Отсюда $d_2 = \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$ см.Теперь найдем длину второй диагонали:$d_1 = 0,8 \cdot 10\sqrt{2} = 8\sqrt{2}$ см.Сторона ромба $a$ связана с его диагоналями через теорему Пифагора, так как диагонали ромба перпендикулярны и делятся точкой пересечения пополам. Сторона является гипотенузой в прямоугольном треугольнике, катетами которого служат половины диагоналей:$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$$a^2 = \left(\frac{8\sqrt{2}}{2}\right)^2 + \left(\frac{10\sqrt{2}}{2}\right)^2 = (4\sqrt{2})^2 + (5\sqrt{2})^2$$a^2 = (16 \cdot 2) + (25 \cdot 2) = 32 + 50 = 82$$a = \sqrt{82}$ см.<br>Ответ: $\sqrt{82}$ см.</p><p><strong>б)</strong>Периметр ромба $P$ равен 2 дм. Так как все стороны ромба равны, длина одной стороны $a$ составляет:$a = \frac{P}{4} = \frac{2 \text{ дм}}{4} = 0,5$ дм.Для удобства вычислений переведем в сантиметры: $a = 5$ см.Диагонали ромба $d_1$ и $d_2$ относятся как 3 : 4. Это можно записать как $d_1 = 3x$ и $d_2 = 4x$ для некоторого коэффициента $x$.Применим теорему Пифагора к треугольнику, образованному стороной ромба и половинами его диагоналей:$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$$5^2 = \left(\frac{3x}{2}\right)^2 + \left(\frac{4x}{2}\right)^2$$25 = (1,5x)^2 + (2x)^2$$25 = 2,25x^2 + 4x^2$$25 = 6,25x^2$$x^2 = \frac{25}{6,25} = 4$$x = \sqrt{4} = 2$.Теперь можем найти длины диагоналей:$d_1 = 3x = 3 \cdot 2 = 6$ см.$d_2 = 4x = 4 \cdot 2 = 8$ см.Площадь ромба $S$ равна половине произведения его диагоналей:$S = \frac{d_1 \cdot d_2}{2} = \frac{6 \cdot 8}{2} = \frac{48}{2} = 24$ см².<br>Ответ: 24 см².</p>" ] #original: array:6 [ "id" => 1553712 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1107808" "type" => "task" ] "text" => "<p><strong>а)</strong>Площадь ромба $S$ вычисляется через его диагонали $d_1$ и $d_2$ по формуле $S = \frac{d_1 \cdot d_2}{2}$. По условию задачи, площадь ромба равна 80 см², а отношение диагоналей $\frac{d_1}{d_2} = 0,8$.Из соотношения диагоналей выразим одну через другую: $d_1 = 0,8 \cdot d_2$.Подставим это выражение в формулу площади:$80 = \frac{(0,8 \cdot d_2) \cdot d_2}{2}$$160 = 0,8 \cdot d_2^2$$d_2^2 = \frac{160}{0,8} = \frac{1600}{8} = 200$Отсюда $d_2 = \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$ см.Теперь найдем длину второй диагонали:$d_1 = 0,8 \cdot 10\sqrt{2} = 8\sqrt{2}$ см.Сторона ромба $a$ связана с его диагоналями через теорему Пифагора, так как диагонали ромба перпендикулярны и делятся точкой пересечения пополам. Сторона является гипотенузой в прямоугольном треугольнике, катетами которого служат половины диагоналей:$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$$a^2 = \left(\frac{8\sqrt{2}}{2}\right)^2 + \left(\frac{10\sqrt{2}}{2}\right)^2 = (4\sqrt{2})^2 + (5\sqrt{2})^2$$a^2 = (16 \cdot 2) + (25 \cdot 2) = 32 + 50 = 82$$a = \sqrt{82}$ см.<br>Ответ: $\sqrt{82}$ см.</p><p><strong>б)</strong>Периметр ромба $P$ равен 2 дм. Так как все стороны ромба равны, длина одной стороны $a$ составляет:$a = \frac{P}{4} = \frac{2 \text{ дм}}{4} = 0,5$ дм.Для удобства вычислений переведем в сантиметры: $a = 5$ см.Диагонали ромба $d_1$ и $d_2$ относятся как 3 : 4. Это можно записать как $d_1 = 3x$ и $d_2 = 4x$ для некоторого коэффициента $x$.Применим теорему Пифагора к треугольнику, образованному стороной ромба и половинами его диагоналей:$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$$5^2 = \left(\frac{3x}{2}\right)^2 + \left(\frac{4x}{2}\right)^2$$25 = (1,5x)^2 + (2x)^2$$25 = 2,25x^2 + 4x^2$$25 = 6,25x^2$$x^2 = \frac{25}{6,25} = 4$$x = \sqrt{4} = 2$.Теперь можем найти длины диагоналей:$d_1 = 3x = 3 \cdot 2 = 6$ см.$d_2 = 4x = 4 \cdot 2 = 8$ см.Площадь ромба $S$ равна половине произведения его диагоналей:$S = \frac{d_1 \cdot d_2}{2} = \frac{6 \cdot 8}{2} = \frac{48}{2} = 24$ см².<br>Ответ: 24 см².</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" => "1107809" "type" => "task" ] "previous" => array:2 [ "refs" => "1107807" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1039 #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" => 1097780 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-38" "field_display_title" => "38" "field_folder" => "folder1" "field_image_name" => "38" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1106 #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 {#1109 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Element {#1129 #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" => 1260931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097780" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1260931 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131} "book_page" => array:2 [ "refs" => "1097780" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1097781" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097779" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:9 [ 0 => App\Models\Task {#1214 #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" => 1107807 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-110" "field_display_title" => "110" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ …1] #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 {#1216 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107808" "type" => "task" ] "previous" => array:2 [ "refs" => "1107806" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097780" "type" => "book_page" ] ] #original: array:24 [ "id" => 1107807 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-110" "field_display_title" => "110" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215} "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 {#1216} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218} "content" => Illuminate\Database\Eloquent\Collection {#1221} "next" => array:2 [ "refs" => "1107808" "type" => "task" ] "previous" => array:2 [ "refs" => "1107806" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227} "page" => array:2 [ "refs" => "1097780" "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 {#1220 #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" => 1107808 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-111" "field_display_title" => "111" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ …1] #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 {#1226 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107809" "type" => "task" ] "previous" => array:2 [ "refs" => "1107807" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097780" "type" => "book_page" ] ] #original: array:24 [ "id" => 1107808 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-111" "field_display_title" => "111" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "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 {#1226} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224} "content" => Illuminate\Database\Eloquent\Collection {#1222} "next" => array:2 [ "refs" => "1107809" "type" => "task" ] "previous" => array:2 [ "refs" => "1107807" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1225} "page" => array:2 [ "refs" => "1097780" "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 {#1223 #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" => 1107809 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-112" "field_display_title" => "112" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ …1] #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 {#1237 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107810" "type" => "task" ] "previous" => array:2 [ "refs" => "1107808" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097780" "type" => "book_page" ] ] #original: array:24 [ "id" => 1107809 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-112" "field_display_title" => "112" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236} "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 {#1237} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1238} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1239} "content" => Illuminate\Database\Eloquent\Collection {#1242} "next" => array:2 [ "refs" => "1107810" "type" => "task" ] "previous" => array:2 [ "refs" => "1107808" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1248} "page" => array:2 [ "refs" => "1097780" "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 {#1241 #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" => 1107810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-113" "field_display_title" => "113" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:1 [ …1] #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 {#1247 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1107811" "type" => "task" ] "previous" => array:2 [ "refs" => "1107809" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097780" "type" => "book_page" ] ] #original: array:24 [ "id" => 1107810 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "38" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-113" "field_display_title" => "113" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1240} "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 {#1247} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245} "content" => Illuminate\Database\Eloquent\Collection {#1244} "next" => array:2 [ "refs" => "1107811" "type" => "task" ] "previous" => array:2 [ "refs" => "1107809" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1262} "page" => array:2 [ "refs" => "1097780" "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 {#1246 #connection: "mysql" #table: "tasks" #primaryKey: 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№111 (с. 38)
Условие. №111 (с. 38)
скриншот условия
111. Найдите:
а) сторону ромба, площадь которого равна $80 \text{ см}^2$, а отношение диагоналей равно 0,8;
б) площадь ромба, периметр которого равен 2 дм, а диагонали относятся как 3 : 4.
Решение. №111 (с. 38)
Решение 2 (rus). №111 (с. 38)
а)Площадь ромба $S$ вычисляется через его диагонали $d_1$ и $d_2$ по формуле $S = \frac{d_1 \cdot d_2}{2}$. По условию задачи, площадь ромба равна 80 см², а отношение диагоналей $\frac{d_1}{d_2} = 0,8$.Из соотношения диагоналей выразим одну через другую: $d_1 = 0,8 \cdot d_2$.Подставим это выражение в формулу площади:$80 = \frac{(0,8 \cdot d_2) \cdot d_2}{2}$$160 = 0,8 \cdot d_2^2$$d_2^2 = \frac{160}{0,8} = \frac{1600}{8} = 200$Отсюда $d_2 = \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$ см.Теперь найдем длину второй диагонали:$d_1 = 0,8 \cdot 10\sqrt{2} = 8\sqrt{2}$ см.Сторона ромба $a$ связана с его диагоналями через теорему Пифагора, так как диагонали ромба перпендикулярны и делятся точкой пересечения пополам. Сторона является гипотенузой в прямоугольном треугольнике, катетами которого служат половины диагоналей:$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$$a^2 = \left(\frac{8\sqrt{2}}{2}\right)^2 + \left(\frac{10\sqrt{2}}{2}\right)^2 = (4\sqrt{2})^2 + (5\sqrt{2})^2$$a^2 = (16 \cdot 2) + (25 \cdot 2) = 32 + 50 = 82$$a = \sqrt{82}$ см.
Ответ: $\sqrt{82}$ см.
б)Периметр ромба $P$ равен 2 дм. Так как все стороны ромба равны, длина одной стороны $a$ составляет:$a = \frac{P}{4} = \frac{2 \text{ дм}}{4} = 0,5$ дм.Для удобства вычислений переведем в сантиметры: $a = 5$ см.Диагонали ромба $d_1$ и $d_2$ относятся как 3 : 4. Это можно записать как $d_1 = 3x$ и $d_2 = 4x$ для некоторого коэффициента $x$.Применим теорему Пифагора к треугольнику, образованному стороной ромба и половинами его диагоналей:$a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2$$5^2 = \left(\frac{3x}{2}\right)^2 + \left(\frac{4x}{2}\right)^2$$25 = (1,5x)^2 + (2x)^2$$25 = 2,25x^2 + 4x^2$$25 = 6,25x^2$$x^2 = \frac{25}{6,25} = 4$$x = \sqrt{4} = 2$.Теперь можем найти длины диагоналей:$d_1 = 3x = 3 \cdot 2 = 6$ см.$d_2 = 4x = 4 \cdot 2 = 8$ см.Площадь ромба $S$ равна половине произведения его диагоналей:$S = \frac{d_1 \cdot d_2}{2} = \frac{6 \cdot 8}{2} = \frac{48}{2} = 24$ см².
Ответ: 24 см².
Другие задания:
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