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Номер 105, страница 37 - 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 105, страница 37.
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Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => 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"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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. 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+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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => 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"field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" 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=> null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #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 {#1095 …2} "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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #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 {#1095 …2} "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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #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 {#1063} "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #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" => 1276460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 …2} "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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "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 {#1102 …2} "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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "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" => "1107802" "type" => "task" ] "text" => "<p><strong>105. а)</strong> Расстояние 300 км пассажирский поезд проходит на 1 ч быстрее товарного. Найдите скорость каждого из поездов, если за 1,5 ч пассажирский поезд проходит на 22,5 км больше, чем товарный.</p><p><strong>б)</strong> Из городов А и В навстречу друг другу одновременно вышли пассажирский и товарный поезда. Двигаясь без остановок с постоянной скоростью, пассажирский поезд прибыл в пункт В через 4 ч, а товарный – в пункт А через 6 ч. Найдите скорость каждого поезда, если через 2 ч после того, как поезда встретились, расстояние между ними составило 320 км.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "105-1.jpg" "alt" => null "width" => "1576" "height" => 625 "path" => "/media/algebra_09/soltan-u19/0-1/105-1.webp?ts=1752834175" ] ] ] #original: array:7 [ "id" => 1276460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1107802" "type" => "task" ] "text" => "<p><strong>105. а)</strong> Расстояние 300 км пассажирский поезд проходит на 1 ч быстрее товарного. Найдите скорость каждого из поездов, если за 1,5 ч пассажирский поезд проходит на 22,5 км больше, чем товарный.</p><p><strong>б)</strong> Из городов А и В навстречу друг другу одновременно вышли пассажирский и товарный поезда. Двигаясь без остановок с постоянной скоростью, пассажирский поезд прибыл в пункт В через 4 ч, а товарный – в пункт А через 6 ч. Найдите скорость каждого поезда, если через 2 ч после того, как поезда встретились, расстояние между ними составило 320 км.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "105-1.jpg" "alt" => null "width" => "1576" "height" => 625 "path" => "/media/algebra_09/soltan-u19/0-1/105-1.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 {#1107 #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" => 1277716 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 …2} "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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "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 {#1110 …2} "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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "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" => "1107802" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "105-1.jpg" "alt" => null "width" => "1420" "height" => 2401 "path" => "/media/algebra_09/soltan-u19/1-1/105-1.webp?ts=1752830836" ] 1 => array:5 [ "name" => "105-2.jpg" "alt" => null "width" => "1249" "height" => 667 "path" => "/media/algebra_09/soltan-u19/1-1/105-2.webp?ts=1752830836" ] ] ] #original: array:6 [ "id" => 1277716 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1107802" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "105-1.jpg" "alt" => null "width" => "1420" "height" => 2401 "path" => "/media/algebra_09/soltan-u19/1-1/105-1.webp?ts=1752830836" ] 1 => array:5 [ "name" => "105-2.jpg" "alt" => null "width" => "1249" "height" => 667 "path" => "/media/algebra_09/soltan-u19/1-1/105-2.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 {#1115 #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" => 1553706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1118 …2} "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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "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 {#1118 …2} "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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "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" => "1107802" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Пусть $v_п$ (км/ч) — скорость пассажирского поезда, а $v_т$ (км/ч) — скорость товарного поезда.</p><p>Время, за которое пассажирский поезд проходит 300 км, равно $t_п = \frac{300}{v_п}$ ч.</p><p>Время, за которое товарный поезд проходит 300 км, равно $t_т = \frac{300}{v_т}$ ч.</p><p>По условию, пассажирский поезд проходит это расстояние на 1 час быстрее, значит:</p><p>$t_т - t_п = 1 \implies \frac{300}{v_т} - \frac{300}{v_п} = 1$</p><p>Также известно, что за 1,5 часа пассажирский поезд проходит на 22,5 км больше, чем товарный. Составим второе уравнение:</p><p>$1.5 \cdot v_п - 1.5 \cdot v_т = 22.5$</p><p>Из второго уравнения выразим разность скоростей:</p><p>$v_п - v_т = \frac{22.5}{1.5} = 15$</p><p>Отсюда $v_п = v_т + 15$. Подставим это выражение в первое уравнение:</p><p>$\frac{300}{v_т} - \frac{300}{v_т + 15} = 1$</p><p>Приведем левую часть к общему знаменателю:</p><p>$\frac{300(v_т + 15) - 300v_т}{v_т(v_т + 15)} = 1$</p><p>$\frac{300v_т + 4500 - 300v_т}{v_т^2 + 15v_т} = 1$</p><p>$\frac{4500}{v_т^2 + 15v_т} = 1$</p><p>$v_т^2 + 15v_т = 4500$</p><p>$v_т^2 + 15v_т - 4500 = 0$</p><p>Решим полученное квадратное уравнение. Найдем дискриминант:</p><p>$D = b^2 - 4ac = 15^2 - 4 \cdot 1 \cdot (-4500) = 225 + 18000 = 18225$</p><p>$\sqrt{D} = \sqrt{18225} = 135$</p><p>Найдем корни уравнения:</p><p>$v_т = \frac{-15 \pm 135}{2}$</p><p>Так как скорость не может быть отрицательной, выбираем корень со знаком «плюс»:</p><p>$v_т = \frac{-15 + 135}{2} = \frac{120}{2} = 60$ (км/ч)</p><p>Теперь найдем скорость пассажирского поезда:</p><p>$v_п = v_т + 15 = 60 + 15 = 75$ (км/ч)</p><p><strong>Ответ:</strong> скорость пассажирского поезда — 75 км/ч, скорость товарного поезда — 60 км/ч.</p><p><strong>б)</strong></p><p>Пусть $v_п$ (км/ч) — скорость пассажирского поезда, $v_т$ (км/ч) — скорость товарного поезда, а $S$ (км) — расстояние между городами А и В.</p><p>Пассажирский поезд, двигаясь из А в В, проходит расстояние $S$ за 4 часа. Следовательно, его скорость $v_п = \frac{S}{4}$.</p><p>Товарный поезд, двигаясь из В в А, проходит расстояние $S$ за 6 часов. Следовательно, его скорость $v_т = \frac{S}{6}$.</p><p>Когда поезда движутся навстречу друг другу, их скорость сближения равна сумме их скоростей $v_{сбл} = v_п + v_т$. После встречи поезда продолжают движение, и их скорость удаления также равна $v_{уд} = v_п + v_т$.</p><p>По условию, через 2 часа после того, как поезда встретились, расстояние между ними составило 320 км. Это расстояние равно произведению скорости удаления на время:</p><p>$(v_п + v_т) \cdot 2 = 320$</p><p>Отсюда находим сумму скоростей поездов:</p><p>$v_п + v_т = \frac{320}{2} = 160$ (км/ч)</p><p>Теперь подставим в это уравнение выражения для скоростей через $S$:</p><p>$\frac{S}{4} + \frac{S}{6} = 160$</p><p>Приведем левую часть к общему знаменателю 12:</p><p>$\frac{3S}{12} + \frac{2S}{12} = 160$</p><p>$\frac{5S}{12} = 160$</p><p>$5S = 160 \cdot 12$</p><p>$S = \frac{160 \cdot 12}{5} = 32 \cdot 12 = 384$ (км)</p><p>Зная расстояние $S$, найдем скорости каждого поезда:</p><p>Скорость пассажирского поезда: $v_п = \frac{S}{4} = \frac{384}{4} = 96$ (км/ч)</p><p>Скорость товарного поезда: $v_т = \frac{S}{6} = \frac{384}{6} = 64$ (км/ч)</p><p><strong>Ответ:</strong> скорость пассажирского поезда — 96 км/ч, скорость товарного поезда — 64 км/ч.</p>" ] #original: array:6 [ "id" => 1553706 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1107802" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Пусть $v_п$ (км/ч) — скорость пассажирского поезда, а $v_т$ (км/ч) — скорость товарного поезда.</p><p>Время, за которое пассажирский поезд проходит 300 км, равно $t_п = \frac{300}{v_п}$ ч.</p><p>Время, за которое товарный поезд проходит 300 км, равно $t_т = \frac{300}{v_т}$ ч.</p><p>По условию, пассажирский поезд проходит это расстояние на 1 час быстрее, значит:</p><p>$t_т - t_п = 1 \implies \frac{300}{v_т} - \frac{300}{v_п} = 1$</p><p>Также известно, что за 1,5 часа пассажирский поезд проходит на 22,5 км больше, чем товарный. Составим второе уравнение:</p><p>$1.5 \cdot v_п - 1.5 \cdot v_т = 22.5$</p><p>Из второго уравнения выразим разность скоростей:</p><p>$v_п - v_т = \frac{22.5}{1.5} = 15$</p><p>Отсюда $v_п = v_т + 15$. Подставим это выражение в первое уравнение:</p><p>$\frac{300}{v_т} - \frac{300}{v_т + 15} = 1$</p><p>Приведем левую часть к общему знаменателю:</p><p>$\frac{300(v_т + 15) - 300v_т}{v_т(v_т + 15)} = 1$</p><p>$\frac{300v_т + 4500 - 300v_т}{v_т^2 + 15v_т} = 1$</p><p>$\frac{4500}{v_т^2 + 15v_т} = 1$</p><p>$v_т^2 + 15v_т = 4500$</p><p>$v_т^2 + 15v_т - 4500 = 0$</p><p>Решим полученное квадратное уравнение. Найдем дискриминант:</p><p>$D = b^2 - 4ac = 15^2 - 4 \cdot 1 \cdot (-4500) = 225 + 18000 = 18225$</p><p>$\sqrt{D} = \sqrt{18225} = 135$</p><p>Найдем корни уравнения:</p><p>$v_т = \frac{-15 \pm 135}{2}$</p><p>Так как скорость не может быть отрицательной, выбираем корень со знаком «плюс»:</p><p>$v_т = \frac{-15 + 135}{2} = \frac{120}{2} = 60$ (км/ч)</p><p>Теперь найдем скорость пассажирского поезда:</p><p>$v_п = v_т + 15 = 60 + 15 = 75$ (км/ч)</p><p><strong>Ответ:</strong> скорость пассажирского поезда — 75 км/ч, скорость товарного поезда — 60 км/ч.</p><p><strong>б)</strong></p><p>Пусть $v_п$ (км/ч) — скорость пассажирского поезда, $v_т$ (км/ч) — скорость товарного поезда, а $S$ (км) — расстояние между городами А и В.</p><p>Пассажирский поезд, двигаясь из А в В, проходит расстояние $S$ за 4 часа. Следовательно, его скорость $v_п = \frac{S}{4}$.</p><p>Товарный поезд, двигаясь из В в А, проходит расстояние $S$ за 6 часов. Следовательно, его скорость $v_т = \frac{S}{6}$.</p><p>Когда поезда движутся навстречу друг другу, их скорость сближения равна сумме их скоростей $v_{сбл} = v_п + v_т$. После встречи поезда продолжают движение, и их скорость удаления также равна $v_{уд} = v_п + v_т$.</p><p>По условию, через 2 часа после того, как поезда встретились, расстояние между ними составило 320 км. Это расстояние равно произведению скорости удаления на время:</p><p>$(v_п + v_т) \cdot 2 = 320$</p><p>Отсюда находим сумму скоростей поездов:</p><p>$v_п + v_т = \frac{320}{2} = 160$ (км/ч)</p><p>Теперь подставим в это уравнение выражения для скоростей через $S$:</p><p>$\frac{S}{4} + \frac{S}{6} = 160$</p><p>Приведем левую часть к общему знаменателю 12:</p><p>$\frac{3S}{12} + \frac{2S}{12} = 160$</p><p>$\frac{5S}{12} = 160$</p><p>$5S = 160 \cdot 12$</p><p>$S = \frac{160 \cdot 12}{5} = 32 \cdot 12 = 384$ (км)</p><p>Зная расстояние $S$, найдем скорости каждого поезда:</p><p>Скорость пассажирского поезда: $v_п = \frac{S}{4} = \frac{384}{4} = 96$ (км/ч)</p><p>Скорость товарного поезда: $v_т = \frac{S}{6} = \frac{384}{6} = 64$ (км/ч)</p><p><strong>Ответ:</strong> скорость пассажирского поезда — 96 км/ч, скорость товарного поезда — 64 км/ч.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] 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"/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 {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "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 } "page" => Illuminate\Database\Eloquent\Collection {#2129 #items: array:1 [ 0 => App\Models\BookPage {#1155 #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" => 1097779 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-37" "field_display_title" => "37" "field_folder" => "folder1" "field_image_name" => "37" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1160 #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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Element {#1190 #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" => 1260930 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260930 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1097780" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097778" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:7 [ 0 => App\Models\Task {#1199 #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" => 1107800 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "36" "field_page_end" => "37" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-103" "field_display_title" => "103" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1330 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107800 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "36" "field_page_end" => "37" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-103" "field_display_title" => "103" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1330 …2} "page" => array:2 [ …2] ] #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 {#1359 #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" => 1107801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-104" "field_display_title" => "104" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1360 …2} "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 {#1362 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1363 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1396 …2} "content" => Illuminate\Database\Eloquent\Collection {#1433 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1458 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107801 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-104" "field_display_title" => "104" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1360 …2} "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 {#1362 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1363 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1396 …2} "content" => Illuminate\Database\Eloquent\Collection {#1433 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1458 …2} "page" => array:2 [ …2] ] #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 {#1487 #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" => 1107802 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-105" "field_display_title" => "105" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1488 …2} "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 {#1490 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1491 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1524 …2} "content" => Illuminate\Database\Eloquent\Collection {#1561 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1586 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107802 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-105" "field_display_title" => "105" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1488 …2} "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 {#1490 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1491 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1524 …2} "content" => Illuminate\Database\Eloquent\Collection {#1561 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1586 …2} "page" => array:2 [ …2] ] #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 {#1615 #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" => 1107803 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-106" "field_display_title" => "106" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1616 …2} "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 {#1618 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1619 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1652 …2} "content" => Illuminate\Database\Eloquent\Collection {#1689 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1714 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107803 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-106" "field_display_title" => "106" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1616 …2} "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 {#1618 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1619 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1652 …2} "content" => Illuminate\Database\Eloquent\Collection {#1689 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1714 …2} "page" => array:2 [ …2] ] #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 {#1743 #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" => 1107804 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-107" "field_display_title" => "107" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1744 …2} "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 {#1746 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1747 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1780 …2} "content" => Illuminate\Database\Eloquent\Collection {#1817 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1842 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107804 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "37" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-107" "field_display_title" => "107" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1744 …2} "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 {#1746 …2} "top_parent_branch" => 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№105 (с. 37)
Условие. №105 (с. 37)
скриншот условия
105. а) Расстояние 300 км пассажирский поезд проходит на 1 ч быстрее товарного. Найдите скорость каждого из поездов, если за 1,5 ч пассажирский поезд проходит на 22,5 км больше, чем товарный.
б) Из городов А и В навстречу друг другу одновременно вышли пассажирский и товарный поезда. Двигаясь без остановок с постоянной скоростью, пассажирский поезд прибыл в пункт В через 4 ч, а товарный – в пункт А через 6 ч. Найдите скорость каждого поезда, если через 2 ч после того, как поезда встретились, расстояние между ними составило 320 км.
Решение. №105 (с. 37)
Решение 2 (rus). №105 (с. 37)
а)
Пусть $v_п$ (км/ч) — скорость пассажирского поезда, а $v_т$ (км/ч) — скорость товарного поезда.
Время, за которое пассажирский поезд проходит 300 км, равно $t_п = \frac{300}{v_п}$ ч.
Время, за которое товарный поезд проходит 300 км, равно $t_т = \frac{300}{v_т}$ ч.
По условию, пассажирский поезд проходит это расстояние на 1 час быстрее, значит:
$t_т - t_п = 1 \implies \frac{300}{v_т} - \frac{300}{v_п} = 1$
Также известно, что за 1,5 часа пассажирский поезд проходит на 22,5 км больше, чем товарный. Составим второе уравнение:
$1.5 \cdot v_п - 1.5 \cdot v_т = 22.5$
Из второго уравнения выразим разность скоростей:
$v_п - v_т = \frac{22.5}{1.5} = 15$
Отсюда $v_п = v_т + 15$. Подставим это выражение в первое уравнение:
$\frac{300}{v_т} - \frac{300}{v_т + 15} = 1$
Приведем левую часть к общему знаменателю:
$\frac{300(v_т + 15) - 300v_т}{v_т(v_т + 15)} = 1$
$\frac{300v_т + 4500 - 300v_т}{v_т^2 + 15v_т} = 1$
$\frac{4500}{v_т^2 + 15v_т} = 1$
$v_т^2 + 15v_т = 4500$
$v_т^2 + 15v_т - 4500 = 0$
Решим полученное квадратное уравнение. Найдем дискриминант:
$D = b^2 - 4ac = 15^2 - 4 \cdot 1 \cdot (-4500) = 225 + 18000 = 18225$
$\sqrt{D} = \sqrt{18225} = 135$
Найдем корни уравнения:
$v_т = \frac{-15 \pm 135}{2}$
Так как скорость не может быть отрицательной, выбираем корень со знаком «плюс»:
$v_т = \frac{-15 + 135}{2} = \frac{120}{2} = 60$ (км/ч)
Теперь найдем скорость пассажирского поезда:
$v_п = v_т + 15 = 60 + 15 = 75$ (км/ч)
Ответ: скорость пассажирского поезда — 75 км/ч, скорость товарного поезда — 60 км/ч.
б)
Пусть $v_п$ (км/ч) — скорость пассажирского поезда, $v_т$ (км/ч) — скорость товарного поезда, а $S$ (км) — расстояние между городами А и В.
Пассажирский поезд, двигаясь из А в В, проходит расстояние $S$ за 4 часа. Следовательно, его скорость $v_п = \frac{S}{4}$.
Товарный поезд, двигаясь из В в А, проходит расстояние $S$ за 6 часов. Следовательно, его скорость $v_т = \frac{S}{6}$.
Когда поезда движутся навстречу друг другу, их скорость сближения равна сумме их скоростей $v_{сбл} = v_п + v_т$. После встречи поезда продолжают движение, и их скорость удаления также равна $v_{уд} = v_п + v_т$.
По условию, через 2 часа после того, как поезда встретились, расстояние между ними составило 320 км. Это расстояние равно произведению скорости удаления на время:
$(v_п + v_т) \cdot 2 = 320$
Отсюда находим сумму скоростей поездов:
$v_п + v_т = \frac{320}{2} = 160$ (км/ч)
Теперь подставим в это уравнение выражения для скоростей через $S$:
$\frac{S}{4} + \frac{S}{6} = 160$
Приведем левую часть к общему знаменателю 12:
$\frac{3S}{12} + \frac{2S}{12} = 160$
$\frac{5S}{12} = 160$
$5S = 160 \cdot 12$
$S = \frac{160 \cdot 12}{5} = 32 \cdot 12 = 384$ (км)
Зная расстояние $S$, найдем скорости каждого поезда:
Скорость пассажирского поезда: $v_п = \frac{S}{4} = \frac{384}{4} = 96$ (км/ч)
Скорость товарного поезда: $v_т = \frac{S}{6} = \frac{384}{6} = 64$ (км/ч)
Ответ: скорость пассажирского поезда — 96 км/ч, скорость товарного поезда — 64 км/ч.
Другие задания:
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