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Номер 98, страница 33 - 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. 2. Системы нелинейных уравнений с двумя переменными - номер 98, страница 33.
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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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} 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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} "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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" => 1107641 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. 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#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" => 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" => "" "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 {#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" => "" "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 {#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] ] #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" => 1107641 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2. Системы нелинейных уравнений с двумя переменными" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "26" "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" => 1276453 "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" => "1107795" "type" => "task" ] "text" => "<p><strong>98.</strong> Решите систему симметрических уравнений:</p><p><strong>a)</strong></p><p>$\begin{cases} x + xy + y = 5, \\ x^2 + xy + y^2 = 7; \end{cases}$</p><p><strong>б)</strong></p><p>$\begin{cases} x + xy + y = 4, \\ x^2 + xy + y^2 = 8. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "98-1.jpg" "alt" => null "width" => "1147" "height" => 309 "path" => "/media/algebra_09/soltan-u19/0-1/98-1.webp?ts=1752834155" ] ] ] #original: array:7 [ "id" => 1276453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1107795" "type" => "task" ] "text" => "<p><strong>98.</strong> Решите систему симметрических уравнений:</p><p><strong>a)</strong></p><p>$\begin{cases} x + xy + y = 5, \\ x^2 + xy + y^2 = 7; \end{cases}$</p><p><strong>б)</strong></p><p>$\begin{cases} x + xy + y = 4, \\ x^2 + xy + y^2 = 8. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "98-1.jpg" "alt" => null "width" => "1147" "height" => 309 "path" => "/media/algebra_09/soltan-u19/0-1/98-1.webp?ts=1752834155" ] ] ] #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" => 1277698 "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" => "1107795" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "98-1.jpg" "alt" => null "width" => "1198" "height" => 1520 "path" => "/media/algebra_09/soltan-u19/1-1/98-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "98-2.jpg" "alt" => null "width" => "1453" "height" => 2393 "path" => "/media/algebra_09/soltan-u19/1-1/98-2.webp?ts=1752830799" ] 2 => array:5 [ "name" => "98-3.jpg" "alt" => null "width" => "1243" "height" => 586 "path" => "/media/algebra_09/soltan-u19/1-1/98-3.webp?ts=1752830799" ] ] ] #original: array:6 [ "id" => 1277698 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1107795" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "98-1.jpg" "alt" => null "width" => "1198" "height" => 1520 "path" => "/media/algebra_09/soltan-u19/1-1/98-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "98-2.jpg" "alt" => null "width" => "1453" "height" => 2393 "path" => "/media/algebra_09/soltan-u19/1-1/98-2.webp?ts=1752830799" ] 2 => array:5 [ "name" => "98-3.jpg" "alt" => null "width" => "1243" "height" => 586 "path" => "/media/algebra_09/soltan-u19/1-1/98-3.webp?ts=1752830799" ] ] ] #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" => 1553699 "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" => "1107795" "type" => "task" ] "text" => "<p><strong>а)</strong>Дана система симметрических уравнений:$ \begin{cases} x + xy + y = 5 \\ x^2 + xy + y^2 = 7 \end{cases} $<br>Для решения введем новые переменные, используя элементарные симметрические многочлены: $u = x + y$ и $v = xy$.<br>Выразим уравнения системы через $u$ и $v$.<br>Первое уравнение: $(x + y) + xy = 5 \implies u + v = 5$.<br>Для второго уравнения преобразуем выражение $x^2 + y^2$. Мы знаем, что $(x+y)^2 = x^2 + 2xy + y^2$, откуда $x^2 + y^2 = (x+y)^2 - 2xy = u^2 - 2v$.<br>Тогда второе уравнение примет вид: $(x^2 + y^2) + xy = 7 \implies (u^2 - 2v) + v = 7 \implies u^2 - v = 7$.<br>Получаем систему уравнений относительно $u$ и $v$:$ \begin{cases} u + v = 5 \\ u^2 - v = 7 \end{cases} $<br>Из первого уравнения выразим $v$: $v = 5 - u$.<br>Подставим это выражение во второе уравнение:$u^2 - (5 - u) = 7$<br>$u^2 + u - 5 = 7$<br>$u^2 + u - 12 = 0$<br>Это квадратное уравнение относительно $u$. Найдем его корни по теореме Виета: $u_1 + u_2 = -1$, $u_1 \cdot u_2 = -12$. Корни: $u_1 = 3$ и $u_2 = -4$.<br>Теперь рассмотрим два случая.<br><em>Случай 1:</em> $u = 3$.<br>Тогда $v = 5 - u = 5 - 3 = 2$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = 3 \\ xy = 2 \end{cases} $<br>Согласно обратной теореме Виета, $x$ и $y$ являются корнями квадратного уравнения $t^2 - 3t + 2 = 0$.<br>Корни этого уравнения: $t_1 = 1, t_2 = 2$.<br>Следовательно, получаем две пары решений: $(1, 2)$ и $(2, 1)$.<br><em>Случай 2:</em> $u = -4$.<br>Тогда $v = 5 - u = 5 - (-4) = 9$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = -4 \\ xy = 9 \end{cases} $<br>$x$ и $y$ являются корнями квадратного уравнения $t^2 - (-4)t + 9 = 0$, то есть $t^2 + 4t + 9 = 0$.<br>Найдем дискриминант этого уравнения: $D = 4^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20$.<br>Поскольку $D < 0$, это уравнение не имеет действительных корней.<br>Таким образом, система имеет только два действительных решения.<br>Ответ: $(1, 2), (2, 1)$.</p><p><strong>б)</strong>Дана система симметрических уравнений:$ \begin{cases} x + xy + y = 4 \\ x^2 + xy + y^2 = 8 \end{cases} $<br>Используем тот же метод замены переменных: $u = x + y$ и $v = xy$.<br>Преобразуем систему:<br>Первое уравнение: $(x + y) + xy = 4 \implies u + v = 4$.<br>Второе уравнение: $(x^2 + y^2) + xy = 8 \implies (u^2 - 2v) + v = 8 \implies u^2 - v = 8$.<br>Получаем систему уравнений относительно $u$ и $v$:$ \begin{cases} u + v = 4 \\ u^2 - v = 8 \end{cases} $<br>Из первого уравнения выразим $v$: $v = 4 - u$.<br>Подставим во второе уравнение:$u^2 - (4 - u) = 8$<br>$u^2 + u - 4 = 8$<br>$u^2 + u - 12 = 0$<br>Корни этого квадратного уравнения: $u_1 = 3$ и $u_2 = -4$.<br>Рассмотрим два случая.<br><em>Случай 1:</em> $u = 3$.<br>Тогда $v = 4 - u = 4 - 3 = 1$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = 3 \\ xy = 1 \end{cases} $<br>$x$ и $y$ являются корнями квадратного уравнения $t^2 - 3t + 1 = 0$.<br>Найдем корни с помощью формулы для корней квадратного уравнения:$t = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}$.<br>Получаем две пары решений: $(\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2})$ и $(\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2})$.<br><em>Случай 2:</em> $u = -4$.<br>Тогда $v = 4 - u = 4 - (-4) = 8$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = -4 \\ xy = 8 \end{cases} $<br>$x$ и $y$ являются корнями уравнения $t^2 - (-4)t + 8 = 0$, то есть $t^2 + 4t + 8 = 0$.<br>Найдем дискриминант: $D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16$.<br>Поскольку $D < 0$, это уравнение не имеет действительных корней.<br>Таким образом, система имеет только два действительных решения.<br>Ответ: $(\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}), (\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2})$.</p>" ] #original: array:6 [ "id" => 1553699 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1107795" "type" => "task" ] "text" => "<p><strong>а)</strong>Дана система симметрических уравнений:$ \begin{cases} x + xy + y = 5 \\ x^2 + xy + y^2 = 7 \end{cases} $<br>Для решения введем новые переменные, используя элементарные симметрические многочлены: $u = x + y$ и $v = xy$.<br>Выразим уравнения системы через $u$ и $v$.<br>Первое уравнение: $(x + y) + xy = 5 \implies u + v = 5$.<br>Для второго уравнения преобразуем выражение $x^2 + y^2$. Мы знаем, что $(x+y)^2 = x^2 + 2xy + y^2$, откуда $x^2 + y^2 = (x+y)^2 - 2xy = u^2 - 2v$.<br>Тогда второе уравнение примет вид: $(x^2 + y^2) + xy = 7 \implies (u^2 - 2v) + v = 7 \implies u^2 - v = 7$.<br>Получаем систему уравнений относительно $u$ и $v$:$ \begin{cases} u + v = 5 \\ u^2 - v = 7 \end{cases} $<br>Из первого уравнения выразим $v$: $v = 5 - u$.<br>Подставим это выражение во второе уравнение:$u^2 - (5 - u) = 7$<br>$u^2 + u - 5 = 7$<br>$u^2 + u - 12 = 0$<br>Это квадратное уравнение относительно $u$. Найдем его корни по теореме Виета: $u_1 + u_2 = -1$, $u_1 \cdot u_2 = -12$. Корни: $u_1 = 3$ и $u_2 = -4$.<br>Теперь рассмотрим два случая.<br><em>Случай 1:</em> $u = 3$.<br>Тогда $v = 5 - u = 5 - 3 = 2$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = 3 \\ xy = 2 \end{cases} $<br>Согласно обратной теореме Виета, $x$ и $y$ являются корнями квадратного уравнения $t^2 - 3t + 2 = 0$.<br>Корни этого уравнения: $t_1 = 1, t_2 = 2$.<br>Следовательно, получаем две пары решений: $(1, 2)$ и $(2, 1)$.<br><em>Случай 2:</em> $u = -4$.<br>Тогда $v = 5 - u = 5 - (-4) = 9$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = -4 \\ xy = 9 \end{cases} $<br>$x$ и $y$ являются корнями квадратного уравнения $t^2 - (-4)t + 9 = 0$, то есть $t^2 + 4t + 9 = 0$.<br>Найдем дискриминант этого уравнения: $D = 4^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20$.<br>Поскольку $D < 0$, это уравнение не имеет действительных корней.<br>Таким образом, система имеет только два действительных решения.<br>Ответ: $(1, 2), (2, 1)$.</p><p><strong>б)</strong>Дана система симметрических уравнений:$ \begin{cases} x + xy + y = 4 \\ x^2 + xy + y^2 = 8 \end{cases} $<br>Используем тот же метод замены переменных: $u = x + y$ и $v = xy$.<br>Преобразуем систему:<br>Первое уравнение: $(x + y) + xy = 4 \implies u + v = 4$.<br>Второе уравнение: $(x^2 + y^2) + xy = 8 \implies (u^2 - 2v) + v = 8 \implies u^2 - v = 8$.<br>Получаем систему уравнений относительно $u$ и $v$:$ \begin{cases} u + v = 4 \\ u^2 - v = 8 \end{cases} $<br>Из первого уравнения выразим $v$: $v = 4 - u$.<br>Подставим во второе уравнение:$u^2 - (4 - u) = 8$<br>$u^2 + u - 4 = 8$<br>$u^2 + u - 12 = 0$<br>Корни этого квадратного уравнения: $u_1 = 3$ и $u_2 = -4$.<br>Рассмотрим два случая.<br><em>Случай 1:</em> $u = 3$.<br>Тогда $v = 4 - u = 4 - 3 = 1$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = 3 \\ xy = 1 \end{cases} $<br>$x$ и $y$ являются корнями квадратного уравнения $t^2 - 3t + 1 = 0$.<br>Найдем корни с помощью формулы для корней квадратного уравнения:$t = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}$.<br>Получаем две пары решений: $(\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2})$ и $(\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2})$.<br><em>Случай 2:</em> $u = -4$.<br>Тогда $v = 4 - u = 4 - (-4) = 8$.<br>Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = -4 \\ xy = 8 \end{cases} $<br>$x$ и $y$ являются корнями уравнения $t^2 - (-4)t + 8 = 0$, то есть $t^2 + 4t + 8 = 0$.<br>Найдем дискриминант: $D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16$.<br>Поскольку $D < 0$, это уравнение не имеет действительных корней.<br>Таким образом, система имеет только два действительных решения.<br>Ответ: $(\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}), (\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2})$.</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" => "1107796" "type" => "task" ] "previous" => array:2 [ "refs" => "1107794" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #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 {#1125 #items: array:1 [ 0 => App\Models\Term {#1126 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Term {#1128 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: 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"field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1133 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" 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Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#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 {#1360 #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" => 1097775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-33" "field_display_title" => "33" "field_folder" => "folder1" "field_image_name" => "33" "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" => 1260926 "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" => 1260926 "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" => "1097776" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097774" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:6 [ 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" => 1107794 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-97" "field_display_title" => "97" "field_outside_task" => null "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 {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1292 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107794 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-97" "field_display_title" => "97" "field_outside_task" => null "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 {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1292 …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 {#1293 #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" => 1107795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-98" "field_display_title" => "98" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1294 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1297 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1305 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107795 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-98" "field_display_title" => "98" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1294 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1297 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1305 …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 {#1306 #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" => 1107796 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-99" "field_display_title" => "99" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1311 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107796 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-99" "field_display_title" => "99" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1311 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …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 {#1319 #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" => 1107797 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-100" "field_display_title" => "100" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320 …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 {#1321 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => Illuminate\Database\Eloquent\Collection {#1324 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1331 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107797 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-100" "field_display_title" => "100" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320 …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 {#1321 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => Illuminate\Database\Eloquent\Collection {#1324 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1331 …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 {#1332 #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" => 1107798 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-101" "field_display_title" => "101" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1333 …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 {#1334 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1335 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1336 …2} "content" => Illuminate\Database\Eloquent\Collection {#1337 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1344 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107798 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-101" "field_display_title" => "101" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1333 …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 {#1334 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1335 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1336 …2} "content" => Illuminate\Database\Eloquent\Collection {#1337 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1344 …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 => "*" ] } 5 => App\Models\Task {#1345 #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" => 1107799 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "33" 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№98 (с. 33)
Условие. №98 (с. 33)
скриншот условия
98. Решите систему симметрических уравнений:
a)
$\begin{cases} x + xy + y = 5, \\ x^2 + xy + y^2 = 7; \end{cases}$
б)
$\begin{cases} x + xy + y = 4, \\ x^2 + xy + y^2 = 8. \end{cases}$
Решение. №98 (с. 33)
Решение 2 (rus). №98 (с. 33)
а)Дана система симметрических уравнений:$ \begin{cases} x + xy + y = 5 \\ x^2 + xy + y^2 = 7 \end{cases} $
Для решения введем новые переменные, используя элементарные симметрические многочлены: $u = x + y$ и $v = xy$.
Выразим уравнения системы через $u$ и $v$.
Первое уравнение: $(x + y) + xy = 5 \implies u + v = 5$.
Для второго уравнения преобразуем выражение $x^2 + y^2$. Мы знаем, что $(x+y)^2 = x^2 + 2xy + y^2$, откуда $x^2 + y^2 = (x+y)^2 - 2xy = u^2 - 2v$.
Тогда второе уравнение примет вид: $(x^2 + y^2) + xy = 7 \implies (u^2 - 2v) + v = 7 \implies u^2 - v = 7$.
Получаем систему уравнений относительно $u$ и $v$:$ \begin{cases} u + v = 5 \\ u^2 - v = 7 \end{cases} $
Из первого уравнения выразим $v$: $v = 5 - u$.
Подставим это выражение во второе уравнение:$u^2 - (5 - u) = 7$
$u^2 + u - 5 = 7$
$u^2 + u - 12 = 0$
Это квадратное уравнение относительно $u$. Найдем его корни по теореме Виета: $u_1 + u_2 = -1$, $u_1 \cdot u_2 = -12$. Корни: $u_1 = 3$ и $u_2 = -4$.
Теперь рассмотрим два случая.
Случай 1: $u = 3$.
Тогда $v = 5 - u = 5 - 3 = 2$.
Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = 3 \\ xy = 2 \end{cases} $
Согласно обратной теореме Виета, $x$ и $y$ являются корнями квадратного уравнения $t^2 - 3t + 2 = 0$.
Корни этого уравнения: $t_1 = 1, t_2 = 2$.
Следовательно, получаем две пары решений: $(1, 2)$ и $(2, 1)$.
Случай 2: $u = -4$.
Тогда $v = 5 - u = 5 - (-4) = 9$.
Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = -4 \\ xy = 9 \end{cases} $
$x$ и $y$ являются корнями квадратного уравнения $t^2 - (-4)t + 9 = 0$, то есть $t^2 + 4t + 9 = 0$.
Найдем дискриминант этого уравнения: $D = 4^2 - 4 \cdot 1 \cdot 9 = 16 - 36 = -20$.
Поскольку $D < 0$, это уравнение не имеет действительных корней.
Таким образом, система имеет только два действительных решения.
Ответ: $(1, 2), (2, 1)$.
б)Дана система симметрических уравнений:$ \begin{cases} x + xy + y = 4 \\ x^2 + xy + y^2 = 8 \end{cases} $
Используем тот же метод замены переменных: $u = x + y$ и $v = xy$.
Преобразуем систему:
Первое уравнение: $(x + y) + xy = 4 \implies u + v = 4$.
Второе уравнение: $(x^2 + y^2) + xy = 8 \implies (u^2 - 2v) + v = 8 \implies u^2 - v = 8$.
Получаем систему уравнений относительно $u$ и $v$:$ \begin{cases} u + v = 4 \\ u^2 - v = 8 \end{cases} $
Из первого уравнения выразим $v$: $v = 4 - u$.
Подставим во второе уравнение:$u^2 - (4 - u) = 8$
$u^2 + u - 4 = 8$
$u^2 + u - 12 = 0$
Корни этого квадратного уравнения: $u_1 = 3$ и $u_2 = -4$.
Рассмотрим два случая.
Случай 1: $u = 3$.
Тогда $v = 4 - u = 4 - 3 = 1$.
Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = 3 \\ xy = 1 \end{cases} $
$x$ и $y$ являются корнями квадратного уравнения $t^2 - 3t + 1 = 0$.
Найдем корни с помощью формулы для корней квадратного уравнения:$t = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}$.
Получаем две пары решений: $(\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2})$ и $(\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2})$.
Случай 2: $u = -4$.
Тогда $v = 4 - u = 4 - (-4) = 8$.
Возвращаемся к переменным $x$ и $y$:$ \begin{cases} x + y = -4 \\ xy = 8 \end{cases} $
$x$ и $y$ являются корнями уравнения $t^2 - (-4)t + 8 = 0$, то есть $t^2 + 4t + 8 = 0$.
Найдем дискриминант: $D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16$.
Поскольку $D < 0$, это уравнение не имеет действительных корней.
Таким образом, система имеет только два действительных решения.
Ответ: $(\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}), (\frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2})$.
Другие задания:
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