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Номер 938, страница 185 - гдз по алгебре 7 класс учебник Мерзляк, Полонский
Авторы: Мерзляк А. Г., Полонский В. Б., Якир М. С.
Тип: Учебник
Издательство: Просвещение, Вентана-граф
Год издания: 2016 - 2022
ISBN: 978-5-360-07440-3
Рекомендовано Министерством образования и науки Российской Федерации
Популярные ГДЗ в 7 классе
Глава 4. Системы линейных уравнений с двумя переменными. §24. Уравнения с двумя переменными - номер 938, страница 185.
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"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 {#1074 #items: array:1 [ 0 => App\Models\Branch {#1034 #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" => 1375460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Системы линейных уравнений с двумя переменными" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "178" 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"field_url" => "/7-klass/algebra/merzlyak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1070 …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" => 864 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2016" "field_publication_year_until" => "2022" "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null 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"field_new_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_url" => "/7-klass/algebra/merzlyak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1106 …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" => 864 "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 {#1078 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_book_type" => 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=> "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1407" "field_priority" => null "field_default_folder" => "/algebra_07/merzlyak/" "field_isbn" => "978-5-360-07440-3" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_url" => "/7-klass/algebra/merzlyak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1106 …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: [] 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"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 {#1076} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1107} ] #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 {#1108 #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" => 1375462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "§24. Уравнения с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "178" "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 {#1110 #items: array:1 [ 0 => App\Models\Book {#1111 #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" => 864 "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 {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2016" "field_publication_year_until" => "2022" "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1407" "field_priority" => null "field_default_folder" => "/algebra_07/merzlyak/" "field_isbn" => "978-5-360-07440-3" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_url" => "/7-klass/algebra/merzlyak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1140 …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" => 864 "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 {#1112 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1136 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2016" "field_publication_year_until" => "2022" "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1407" "field_priority" => null "field_default_folder" => "/algebra_07/merzlyak/" "field_isbn" => "978-5-360-07440-3" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} …9 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\Branch {#1142 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1375462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "§24. Уравнения с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1109} "field_page_start" => "178" "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 {#1110} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1141} ] #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 {#1180 #items: array:7 [ 0 => App\Models\Element {#1193 #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" => 1605893 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Edition {#1194 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "text" => "<p><strong>938.</strong> Ученикам 7 класса на математическом конкурсе предложили решить задачи по алгебре и по геометрии. За каждую правильно решённую задачу по алгебре начислялось 2 балла, а за задачу по геометрии – 3 балла. Максимальное количество набранных баллов могло составить 24.</p><p>Сколько было предложено задач отдельно по алгебре и по геометрии, если по каждому из этих предметов была хотя бы одна задача? Найдите все возможные ответы.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "536" "height" => 166 "path" => "/media/algebra_07/merzlyak/0-00/938-1.webp?ts=1757507358" ] ] ] #original: array:7 [ "id" => 1605893 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1202} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "text" => "<p><strong>938.</strong> Ученикам 7 класса на математическом конкурсе предложили решить задачи по алгебре и по геометрии. За каждую правильно решённую задачу по алгебре начислялось 2 балла, а за задачу по геометрии – 3 балла. Максимальное количество набранных баллов могло составить 24.</p><p>Сколько было предложено задач отдельно по алгебре и по геометрии, если по каждому из этих предметов была хотя бы одна задача? Найдите все возможные ответы.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "536" "height" => 166 "path" => "/media/algebra_07/merzlyak/0-00/938-1.webp?ts=1757507358" ] ] ] #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 {#1232 #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" => 1609236 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ 0 => App\Models\Edition {#1233 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.png" "alt" => null "width" => "720" "height" => 3044 "path" => "/media/algebra_07/merzlyak/1-00/938-1.webp?ts=1757508797" ] ] ] #original: array:6 [ "id" => 1609236 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1242} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.png" "alt" => null "width" => "720" "height" => 3044 "path" => "/media/algebra_07/merzlyak/1-00/938-1.webp?ts=1757508797" ] ] ] #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 {#1200 #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" => 1606379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Edition {#1201 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "660" "height" => 636 "path" => "/media/algebra_07/merzlyak/2-00/938-1.webp?ts=1757507390" ] ] ] #original: array:6 [ "id" => 1606379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1210} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "660" "height" => 636 "path" => "/media/algebra_07/merzlyak/2-00/938-1.webp?ts=1757507390" ] ] ] #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\Element {#1208 #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" => 1606460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Edition {#1209 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.png" "alt" => null "width" => "1400" "height" => 1754 "path" => "/media/algebra_07/merzlyak/3-00/938-1.webp?ts=1757507417" ] ] ] #original: array:6 [ "id" => 1606460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1218} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.png" "alt" => null "width" => "1400" "height" => 1754 "path" => "/media/algebra_07/merzlyak/3-00/938-1.webp?ts=1757507417" ] ] ] #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\Element {#1224 #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" => 1607399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:1 [ 0 => App\Models\Edition {#1225 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "817" "height" => 489 "path" => "/media/algebra_07/merzlyak/4-00/938-1.webp?ts=1757507399" ] ] ] #original: array:6 [ "id" => 1607399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1234} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "817" "height" => 489 "path" => "/media/algebra_07/merzlyak/4-00/938-1.webp?ts=1757507399" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1216 #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" => 1606785 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Edition {#1217 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "600" "height" => 787 "path" => "/media/algebra_07/merzlyak/5-00/938-1.webp?ts=1757507366" ] ] ] #original: array:6 [ "id" => 1606785 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1226} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "938-1.jpg" "alt" => null "width" => "600" "height" => 787 "path" => "/media/algebra_07/merzlyak/5-00/938-1.webp?ts=1757507366" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Element {#1240 #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" => 1794011 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ 0 => App\Models\Edition {#1241 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "text" => "<p>Пусть $x$ — количество задач по алгебре, а $y$ — количество задач по геометрии.</p><p>За каждую правильно решённую задачу по алгебре начислялось 2 балла, а за задачу по геометрии — 3 балла. Максимальное количество баллов, которое можно было набрать, равно 24. Это означает, что сумма баллов за все задачи по алгебре и все задачи по геометрии равна 24. Составим уравнение на основе этих данных:</p><p>$2x + 3y = 24$</p><p>По условию задачи, было предложено хотя бы по одной задаче по каждому из предметов. Это означает, что $x$ и $y$ должны быть натуральными числами, то есть $x \ge 1$ и $y \ge 1$.</p><p>Нам необходимо найти все пары натуральных чисел $(x, y)$, которые являются решением данного уравнения. Такое уравнение называется диофантовым.</p><p>Выразим одну переменную через другую. Удобнее выразить $x$:</p><p>$2x = 24 - 3y$</p><p>$x = \frac{24 - 3y}{2}$</p><p>$x = 12 - \frac{3}{2}y$</p><p>Поскольку $x$ должно быть целым числом, то и выражение $12 - \frac{3}{2}y$ должно быть целым. Так как 12 — целое число, то дробь $\frac{3}{2}y$ также должна давать в результате целое число. Это возможно только в том случае, если $y$ является чётным числом, так как числитель 3 не делится на знаменатель 2.</p><p>Кроме того, мы знаем, что $x \ge 1$. Подставим выражение для $x$ в это неравенство:</p><p>$12 - \frac{3}{2}y \ge 1$</p><p>Вычтем 12 из обеих частей:</p><p>$-\frac{3}{2}y \ge -11$</p><p>Умножим обе части на -1, изменив знак неравенства на противоположный:</p><p>$\frac{3}{2}y \le 11$</p><p>$3y \le 22$</p><p>$y \le \frac{22}{3}$</p><p>$y \le 7\frac{1}{3}$</p><p>Таким образом, мы ищем значения $y$, которые являются чётными натуральными числами и не превышают $7\frac{1}{3}$. Этим условиям удовлетворяют числа: 2, 4, 6.</p><p>Теперь найдём соответствующие значения $x$ для каждого возможного значения $y$:</p><p>1. Если $y = 2$:<br>$x = 12 - \frac{3 \cdot 2}{2} = 12 - 3 = 9$.<br>Эта пара ($x=9, y=2$) удовлетворяет всем условиям. Значит, могло быть 9 задач по алгебре и 2 по геометрии.</p><p>2. Если $y = 4$:<br>$x = 12 - \frac{3 \cdot 4}{2} = 12 - 6 = 6$.<br>Эта пара ($x=6, y=4$) также удовлетворяет всем условиям. Значит, могло быть 6 задач по алгебре и 4 по геометрии.</p><p>3. Если $y = 6$:<br>$x = 12 - \frac{3 \cdot 6}{2} = 12 - 9 = 3$.<br>Эта пара ($x=3, y=6$) тоже удовлетворяет всем условиям. Значит, могло быть 3 задачи по алгебре и 6 по геометрии.</p><p>Если взять следующее чётное число $y=8$, то $x = 12 - \frac{3 \cdot 8}{2} = 12 - 12 = 0$, что не удовлетворяет условию $x \ge 1$.</p><p>Следовательно, существует три возможных варианта.</p><p>Ответ: существует три возможных варианта: 9 задач по алгебре и 2 задачи по геометрии; или 6 задач по алгебре и 4 задачи по геометрии; или 3 задачи по алгебре и 6 задач по геометрии.</p>" ] #original: array:6 [ "id" => 1794011 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1250} "task" => array:2 [ "refs" => "1376559" "type" => "task" ] "text" => "<p>Пусть $x$ — количество задач по алгебре, а $y$ — количество задач по геометрии.</p><p>За каждую правильно решённую задачу по алгебре начислялось 2 балла, а за задачу по геометрии — 3 балла. Максимальное количество баллов, которое можно было набрать, равно 24. Это означает, что сумма баллов за все задачи по алгебре и все задачи по геометрии равна 24. Составим уравнение на основе этих данных:</p><p>$2x + 3y = 24$</p><p>По условию задачи, было предложено хотя бы по одной задаче по каждому из предметов. Это означает, что $x$ и $y$ должны быть натуральными числами, то есть $x \ge 1$ и $y \ge 1$.</p><p>Нам необходимо найти все пары натуральных чисел $(x, y)$, которые являются решением данного уравнения. Такое уравнение называется диофантовым.</p><p>Выразим одну переменную через другую. Удобнее выразить $x$:</p><p>$2x = 24 - 3y$</p><p>$x = \frac{24 - 3y}{2}$</p><p>$x = 12 - \frac{3}{2}y$</p><p>Поскольку $x$ должно быть целым числом, то и выражение $12 - \frac{3}{2}y$ должно быть целым. Так как 12 — целое число, то дробь $\frac{3}{2}y$ также должна давать в результате целое число. Это возможно только в том случае, если $y$ является чётным числом, так как числитель 3 не делится на знаменатель 2.</p><p>Кроме того, мы знаем, что $x \ge 1$. Подставим выражение для $x$ в это неравенство:</p><p>$12 - \frac{3}{2}y \ge 1$</p><p>Вычтем 12 из обеих частей:</p><p>$-\frac{3}{2}y \ge -11$</p><p>Умножим обе части на -1, изменив знак неравенства на противоположный:</p><p>$\frac{3}{2}y \le 11$</p><p>$3y \le 22$</p><p>$y \le \frac{22}{3}$</p><p>$y \le 7\frac{1}{3}$</p><p>Таким образом, мы ищем значения $y$, которые являются чётными натуральными числами и не превышают $7\frac{1}{3}$. Этим условиям удовлетворяют числа: 2, 4, 6.</p><p>Теперь найдём соответствующие значения $x$ для каждого возможного значения $y$:</p><p>1. Если $y = 2$:<br>$x = 12 - \frac{3 \cdot 2}{2} = 12 - 3 = 9$.<br>Эта пара ($x=9, y=2$) удовлетворяет всем условиям. Значит, могло быть 9 задач по алгебре и 2 по геометрии.</p><p>2. Если $y = 4$:<br>$x = 12 - \frac{3 \cdot 4}{2} = 12 - 6 = 6$.<br>Эта пара ($x=6, y=4$) также удовлетворяет всем условиям. Значит, могло быть 6 задач по алгебре и 4 по геометрии.</p><p>3. Если $y = 6$:<br>$x = 12 - \frac{3 \cdot 6}{2} = 12 - 9 = 3$.<br>Эта пара ($x=3, y=6$) тоже удовлетворяет всем условиям. Значит, могло быть 3 задачи по алгебре и 6 по геометрии.</p><p>Если взять следующее чётное число $y=8$, то $x = 12 - \frac{3 \cdot 8}{2} = 12 - 12 = 0$, что не удовлетворяет условию $x \ge 1$.</p><p>Следовательно, существует три возможных варианта.</p><p>Ответ: существует три возможных варианта: 9 задач по алгебре и 2 задачи по геометрии; или 6 задач по алгебре и 4 задачи по геометрии; или 3 задачи по алгебре и 6 задач по геометрии.</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" => "1376560" "type" => "task" ] "previous" => array:2 [ "refs" => "1376558" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ 0 => App\Models\Book {#1191 #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" => 864 "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 {#1179 #items: array:1 [ 0 => App\Models\Term {#1178 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1190 #items: array:1 [ 0 => App\Models\Term {#1189 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:2 [ 0 => App\Models\Term {#1186 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1185 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:3 [ 0 => App\Models\Term {#1183 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1181 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1184 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1248 #items: array:1 [ 0 => App\Models\Term {#1249 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Term {#1252 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:1 [ 0 => App\Models\Term {#1254 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1255 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1256 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1257 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:1 [ 0 => App\Models\Term {#1259 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1260 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1261 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2016" "field_publication_year_until" => "2022" "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1407" "field_priority" => null "field_default_folder" => "/algebra_07/merzlyak/" "field_isbn" => "978-5-360-07440-3" "field_cover" => array:1 [ 0 => "/media/algebra_07/merzlyak/covers/cover1.webp?ts=1757498550" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_07/merzlyak/covers/cover1.webp?ts=1757498550" "alt" => "" "width" => "600" "height" => "780" ] ] "field_popular_book" => "0" "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1262 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1263 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1264 #items: [] #escapeWhenCastingToString: false } "field_url" => "/7-klass/algebra/merzlyak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection 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"field_series" => Illuminate\Database\Eloquent\Collection {#1255} "field_umk" => Illuminate\Database\Eloquent\Collection {#1256} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1257} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1258} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1260} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1261} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2016" "field_publication_year_until" => "2022" "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1407" "field_priority" => null 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№938 (с. 185)
Условие. №938 (с. 185)
скриншот условия
938. Ученикам 7 класса на математическом конкурсе предложили решить задачи по алгебре и по геометрии. За каждую правильно решённую задачу по алгебре начислялось 2 балла, а за задачу по геометрии – 3 балла. Максимальное количество набранных баллов могло составить 24.
Сколько было предложено задач отдельно по алгебре и по геометрии, если по каждому из этих предметов была хотя бы одна задача? Найдите все возможные ответы.
Решение 1. №938 (с. 185)
Решение 2. №938 (с. 185)
Решение 3. №938 (с. 185)
Решение 4. №938 (с. 185)
Решение 5. №938 (с. 185)
Решение 6. №938 (с. 185)
Пусть $x$ — количество задач по алгебре, а $y$ — количество задач по геометрии.
За каждую правильно решённую задачу по алгебре начислялось 2 балла, а за задачу по геометрии — 3 балла. Максимальное количество баллов, которое можно было набрать, равно 24. Это означает, что сумма баллов за все задачи по алгебре и все задачи по геометрии равна 24. Составим уравнение на основе этих данных:
$2x + 3y = 24$
По условию задачи, было предложено хотя бы по одной задаче по каждому из предметов. Это означает, что $x$ и $y$ должны быть натуральными числами, то есть $x \ge 1$ и $y \ge 1$.
Нам необходимо найти все пары натуральных чисел $(x, y)$, которые являются решением данного уравнения. Такое уравнение называется диофантовым.
Выразим одну переменную через другую. Удобнее выразить $x$:
$2x = 24 - 3y$
$x = \frac{24 - 3y}{2}$
$x = 12 - \frac{3}{2}y$
Поскольку $x$ должно быть целым числом, то и выражение $12 - \frac{3}{2}y$ должно быть целым. Так как 12 — целое число, то дробь $\frac{3}{2}y$ также должна давать в результате целое число. Это возможно только в том случае, если $y$ является чётным числом, так как числитель 3 не делится на знаменатель 2.
Кроме того, мы знаем, что $x \ge 1$. Подставим выражение для $x$ в это неравенство:
$12 - \frac{3}{2}y \ge 1$
Вычтем 12 из обеих частей:
$-\frac{3}{2}y \ge -11$
Умножим обе части на -1, изменив знак неравенства на противоположный:
$\frac{3}{2}y \le 11$
$3y \le 22$
$y \le \frac{22}{3}$
$y \le 7\frac{1}{3}$
Таким образом, мы ищем значения $y$, которые являются чётными натуральными числами и не превышают $7\frac{1}{3}$. Этим условиям удовлетворяют числа: 2, 4, 6.
Теперь найдём соответствующие значения $x$ для каждого возможного значения $y$:
1. Если $y = 2$:
$x = 12 - \frac{3 \cdot 2}{2} = 12 - 3 = 9$.
Эта пара ($x=9, y=2$) удовлетворяет всем условиям. Значит, могло быть 9 задач по алгебре и 2 по геометрии.
2. Если $y = 4$:
$x = 12 - \frac{3 \cdot 4}{2} = 12 - 6 = 6$.
Эта пара ($x=6, y=4$) также удовлетворяет всем условиям. Значит, могло быть 6 задач по алгебре и 4 по геометрии.
3. Если $y = 6$:
$x = 12 - \frac{3 \cdot 6}{2} = 12 - 9 = 3$.
Эта пара ($x=3, y=6$) тоже удовлетворяет всем условиям. Значит, могло быть 3 задачи по алгебре и 6 по геометрии.
Если взять следующее чётное число $y=8$, то $x = 12 - \frac{3 \cdot 8}{2} = 12 - 12 = 0$, что не удовлетворяет условию $x \ge 1$.
Следовательно, существует три возможных варианта.
Ответ: существует три возможных варианта: 9 задач по алгебре и 2 задачи по геометрии; или 6 задач по алгебре и 4 задачи по геометрии; или 3 задачи по алгебре и 6 задач по геометрии.
Другие задания:
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