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Номер 192, страница 57 - 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. 6. Упражнения на повторение раздела «Уравнения, неравенства с двумя переменными и их системы» - номер 192, страница 57.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107893 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-192" "field_display_title" => "192" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #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" => 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 {#1039 #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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …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 {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. 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Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #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 {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #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 {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …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 {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …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 {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1101 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107639 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Уравнения, неравенства с двумя переменными и их системы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "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 {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #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 {#1102 #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" => 1107645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "6. Упражнения на повторение раздела «Уравнения, неравенства с двумя переменными и их системы»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "54" "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 {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #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 {#1135 …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 {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …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 {#1135 …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 {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …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" => 1107645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "6. Упражнения на повторение раздела «Уравнения, неравенства с двумя переменными и их системы»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "54" "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 {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1148 #items: array:3 [ 0 => App\Models\Element {#1158 #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" => 1276547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Edition {#1159 #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 {#1161 …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 {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …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 {#1161 …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 {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …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" => "1107893" "type" => "task" ] "text" => "<p><strong>192.</strong> Решите графически систему уравнений:</p><p><strong>а)</strong> $\begin{cases} y = |x^2 - 6x + 5|, \\ y - x + 1 = 0; \end{cases}$ <strong>б)</strong> $\begin{cases} x^2 + y^2 - 2y = 8, \\ xy + y = 4. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "192-1.jpg" "alt" => null "width" => "1093" "height" => 292 "path" => "/media/algebra_09/soltan-u19/0-1/192-1.webp?ts=1752834240" ] ] ] #original: array:7 [ "id" => 1276547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167} "task" => array:2 [ "refs" => "1107893" "type" => "task" ] "text" => "<p><strong>192.</strong> Решите графически систему уравнений:</p><p><strong>а)</strong> $\begin{cases} y = |x^2 - 6x + 5|, \\ y - x + 1 = 0; \end{cases}$ <strong>б)</strong> $\begin{cases} x^2 + y^2 - 2y = 8, \\ xy + y = 4. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "192-1.jpg" "alt" => null "width" => "1093" "height" => 292 "path" => "/media/algebra_09/soltan-u19/0-1/192-1.webp?ts=1752834240" ] ] ] #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 {#1165 #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" => 1277838 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1166 #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 {#1169 …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 {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …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 {#1169 …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 {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …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" => "1107893" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "192-1.jpg" "alt" => null "width" => "1068" "height" => 1997 "path" => "/media/algebra_09/soltan-u19/1-1/192-1.webp?ts=1752830923" ] ] ] #original: array:6 [ "id" => 1277838 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1107893" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "192-1.jpg" "alt" => null "width" => "1068" "height" => 1997 "path" => "/media/algebra_09/soltan-u19/1-1/192-1.webp?ts=1752830923" ] ] ] #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 {#1173 #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" => 1553793 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Edition {#1174 #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 {#1177 …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 {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …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 {#1177 …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 {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …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" => "1107893" "type" => "task" ] "text" => "<p><strong>а)</strong> Решим систему уравнений $\{_{y - x + 1 = 0}^{y = |x^2 - 6x + 5|}\}$ графически.<br>1. Построим график функции $y = |x^2 - 6x + 5|$.<br>Для этого сначала построим параболу $y = x^2 - 6x + 5$. Это квадратичная функция, графиком которой является парабола с ветвями, направленными вверх.<br>Найдем координаты вершины параболы по формуле $x_0 = -b/(2a)$:<br>$x_0 = -(-6)/(2 \cdot 1) = 3$.<br>$y_0 = 3^2 - 6 \cdot 3 + 5 = 9 - 18 + 5 = -4$.<br>Вершина параболы находится в точке $(3, -4)$.<br>Найдем нули функции (точки пересечения с осью Ox), решив уравнение $x^2 - 6x + 5 = 0$. По теореме Виета, корни уравнения $x_1 = 1$ и $x_2 = 5$.<br>График функции $y = |x^2 - 6x + 5|$ получается из графика параболы $y = x^2 - 6x + 5$ путем симметричного отражения относительно оси Ox той части графика, которая находится ниже оси Ox (на интервале $x \in (1, 5)$). При этом вершина $(3, -4)$ перейдет в точку $(3, 4)$.<br>2. Построим график уравнения $y - x + 1 = 0$.<br>Выразим $y$: $y = x - 1$. Это линейная функция, графиком которой является прямая.<br>Для построения прямой найдем координаты двух точек:<br>При $x = 1$, $y = 1 - 1 = 0$. Точка $(1, 0)$.<br>При $x = 6$, $y = 6 - 1 = 5$. Точка $(6, 5)$.<br>3. Построим оба графика в одной системе координат. Точки пересечения графиков являются решениями системы.<br>Графики пересекаются в трех точках. Из построения видно, что это точки с координатами $(1, 0)$, $(4, 3)$ и $(6, 5)$.<br>Ответ: $(1, 0)$, $(4, 3)$, $(6, 5)$.</p><p><strong>б)</strong> Решим систему уравнений $\{_{xy + y = 4}^{x^2 + y^2 - 2y = 8}\}$ графически.<br>1. Проанализируем и построим график первого уравнения: $x^2 + y^2 - 2y = 8$.<br>Преобразуем уравнение, выделив полный квадрат относительно переменной $y$:<br>$x^2 + (y^2 - 2y + 1) - 1 = 8$<br>$x^2 + (y - 1)^2 = 9$<br>$x^2 + (y - 1)^2 = 3^2$<br>Это стандартное уравнение окружности с центром в точке $C(0, 1)$ и радиусом $R = 3$.<br>2. Проанализируем и построим график второго уравнения: $xy + y = 4$.<br>Вынесем $y$ за скобки: $y(x + 1) = 4$.<br>Если $x \neq -1$, то $y = \frac{4}{x+1}$.<br>Графиком этой функции является гипербола. Ее можно получить сдвигом графика функции $y = 4/x$ на 1 единицу влево вдоль оси Ox. Асимптоты гиперболы: вертикальная $x = -1$ и горизонтальная $y = 0$.<br>3. Построим окружность и гиперболу в одной системе координат. Координаты точек пересечения графиков будут решениями системы.<br>Из графиков видно, что они пересекаются в двух точках. Определим их координаты.<br>Первая точка пересечения имеет координаты $(3, 1)$.<br>Вторая точка пересечения имеет координаты $(0, 4)$.<br>Выполним проверку для найденных точек:<br>Для точки $(3, 1)$: $3^2 + (1-1)^2 = 9+0=9$ и $3 \cdot 1 + 1 = 4$. Оба равенства верны.<br>Для точки $(0, 4)$: $0^2 + (4-1)^2 = 0+9=9$ и $0 \cdot 4 + 4 = 4$. Оба равенства верны.<br>Таким образом, решениями системы являются две пары чисел.<br>Ответ: $(3, 1)$, $(0, 4)$.</p>" ] #original: array:6 [ "id" => 1553793 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183} "task" => array:2 [ "refs" => "1107893" "type" => "task" ] "text" => "<p><strong>а)</strong> Решим систему уравнений $\{_{y - x + 1 = 0}^{y = |x^2 - 6x + 5|}\}$ графически.<br>1. Построим график функции $y = |x^2 - 6x + 5|$.<br>Для этого сначала построим параболу $y = x^2 - 6x + 5$. Это квадратичная функция, графиком которой является парабола с ветвями, направленными вверх.<br>Найдем координаты вершины параболы по формуле $x_0 = -b/(2a)$:<br>$x_0 = -(-6)/(2 \cdot 1) = 3$.<br>$y_0 = 3^2 - 6 \cdot 3 + 5 = 9 - 18 + 5 = -4$.<br>Вершина параболы находится в точке $(3, -4)$.<br>Найдем нули функции (точки пересечения с осью Ox), решив уравнение $x^2 - 6x + 5 = 0$. По теореме Виета, корни уравнения $x_1 = 1$ и $x_2 = 5$.<br>График функции $y = |x^2 - 6x + 5|$ получается из графика параболы $y = x^2 - 6x + 5$ путем симметричного отражения относительно оси Ox той части графика, которая находится ниже оси Ox (на интервале $x \in (1, 5)$). При этом вершина $(3, -4)$ перейдет в точку $(3, 4)$.<br>2. Построим график уравнения $y - x + 1 = 0$.<br>Выразим $y$: $y = x - 1$. Это линейная функция, графиком которой является прямая.<br>Для построения прямой найдем координаты двух точек:<br>При $x = 1$, $y = 1 - 1 = 0$. Точка $(1, 0)$.<br>При $x = 6$, $y = 6 - 1 = 5$. Точка $(6, 5)$.<br>3. Построим оба графика в одной системе координат. Точки пересечения графиков являются решениями системы.<br>Графики пересекаются в трех точках. Из построения видно, что это точки с координатами $(1, 0)$, $(4, 3)$ и $(6, 5)$.<br>Ответ: $(1, 0)$, $(4, 3)$, $(6, 5)$.</p><p><strong>б)</strong> Решим систему уравнений $\{_{xy + y = 4}^{x^2 + y^2 - 2y = 8}\}$ графически.<br>1. Проанализируем и построим график первого уравнения: $x^2 + y^2 - 2y = 8$.<br>Преобразуем уравнение, выделив полный квадрат относительно переменной $y$:<br>$x^2 + (y^2 - 2y + 1) - 1 = 8$<br>$x^2 + (y - 1)^2 = 9$<br>$x^2 + (y - 1)^2 = 3^2$<br>Это стандартное уравнение окружности с центром в точке $C(0, 1)$ и радиусом $R = 3$.<br>2. Проанализируем и построим график второго уравнения: $xy + y = 4$.<br>Вынесем $y$ за скобки: $y(x + 1) = 4$.<br>Если $x \neq -1$, то $y = \frac{4}{x+1}$.<br>Графиком этой функции является гипербола. Ее можно получить сдвигом графика функции $y = 4/x$ на 1 единицу влево вдоль оси Ox. Асимптоты гиперболы: вертикальная $x = -1$ и горизонтальная $y = 0$.<br>3. Построим окружность и гиперболу в одной системе координат. Координаты точек пересечения графиков будут решениями системы.<br>Из графиков видно, что они пересекаются в двух точках. Определим их координаты.<br>Первая точка пересечения имеет координаты $(3, 1)$.<br>Вторая точка пересечения имеет координаты $(0, 4)$.<br>Выполним проверку для найденных точек:<br>Для точки $(3, 1)$: $3^2 + (1-1)^2 = 9+0=9$ и $3 \cdot 1 + 1 = 4$. Оба равенства верны.<br>Для точки $(0, 4)$: $0^2 + (4-1)^2 = 0+9=9$ и $0 \cdot 4 + 4 = 4$. Оба равенства верны.<br>Таким образом, решениями системы являются две пары чисел.<br>Ответ: $(3, 1)$, $(0, 4)$.</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" => "1107894" "type" => "task" ] "previous" => array:2 [ "refs" => "1107892" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1155 #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 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"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 {#1139 #items: array:1 [ 0 => App\Models\Term {#1146 #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" 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"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: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1142 #items: array:3 [ 0 => App\Models\Term {#1143 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1149 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false 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"field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Term {#1153 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false 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+preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #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 {#1141} "field_class" => Illuminate\Database\Eloquent\Collection {#1139} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145} "field_author" => Illuminate\Database\Eloquent\Collection {#1142} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1154} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_country" => Illuminate\Database\Eloquent\Collection {#1150} "field_city" => Illuminate\Database\Eloquent\Collection {#1151} "field_series" => Illuminate\Database\Eloquent\Collection {#1182} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "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 {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1489 #items: array:1 [ 0 => App\Models\BookPage {#1032 #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" => 1097799 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-57" "field_display_title" => "57" "field_folder" => "folder1" "field_image_name" => "57" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#1193 #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 {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1217 …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 {#1218 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1217 …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 {#1218 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1222 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => 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" => 1260950 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260950 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1225 …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" => "1097800" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097798" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1232 #items: array:7 [ 0 => App\Models\Task {#1233 #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" => 1107889 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-188" "field_display_title" => "188" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1234 …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 {#1236 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 …2} "content" => Illuminate\Database\Eloquent\Collection {#1335 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1360 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107889 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-188" "field_display_title" => "188" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1234 …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 {#1236 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1237 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 …2} "content" => Illuminate\Database\Eloquent\Collection {#1335 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1360 …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 {#1361 #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" => 1107890 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-189" "field_display_title" => "189" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1362 …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 {#1363 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1364 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1365 …2} "content" => Illuminate\Database\Eloquent\Collection {#1366 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1391 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107890 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-189" "field_display_title" => "189" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1362 …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 {#1363 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1364 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1365 …2} "content" => Illuminate\Database\Eloquent\Collection {#1366 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1391 …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 {#1392 #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" => 1107891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-190" "field_display_title" => "190" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1393 …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 {#1394 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1395 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1396 …2} "content" => Illuminate\Database\Eloquent\Collection {#1397 …2} …4 ] #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] } 3 => App\Models\Task {#1423 #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 [ …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] } 4 => App\Models\Task {#1448 #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 [ …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] } 5 => App\Models\Task {#1461 #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 [ …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] } 6 => App\Models\Task {#1474 #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 [ …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:21 [ "id" => 1097799 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "57" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-57" "field_display_title" => "57" "field_folder" => "folder1" "field_image_name" => "57" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026} "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 {#1222} "content" => Illuminate\Database\Eloquent\Collection {#1223} "next" => array:2 [ "refs" => "1097800" "type" => "book_page" ] "previous" => array:2 [ "refs" => 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№192 (с. 57)
Условие. №192 (с. 57)
скриншот условия
192. Решите графически систему уравнений:
а) $\begin{cases} y = |x^2 - 6x + 5|, \\ y - x + 1 = 0; \end{cases}$ б) $\begin{cases} x^2 + y^2 - 2y = 8, \\ xy + y = 4. \end{cases}$
Решение. №192 (с. 57)
Решение 2 (rus). №192 (с. 57)
а) Решим систему уравнений $\{_{y - x + 1 = 0}^{y = |x^2 - 6x + 5|}\}$ графически.
1. Построим график функции $y = |x^2 - 6x + 5|$.
Для этого сначала построим параболу $y = x^2 - 6x + 5$. Это квадратичная функция, графиком которой является парабола с ветвями, направленными вверх.
Найдем координаты вершины параболы по формуле $x_0 = -b/(2a)$:
$x_0 = -(-6)/(2 \cdot 1) = 3$.
$y_0 = 3^2 - 6 \cdot 3 + 5 = 9 - 18 + 5 = -4$.
Вершина параболы находится в точке $(3, -4)$.
Найдем нули функции (точки пересечения с осью Ox), решив уравнение $x^2 - 6x + 5 = 0$. По теореме Виета, корни уравнения $x_1 = 1$ и $x_2 = 5$.
График функции $y = |x^2 - 6x + 5|$ получается из графика параболы $y = x^2 - 6x + 5$ путем симметричного отражения относительно оси Ox той части графика, которая находится ниже оси Ox (на интервале $x \in (1, 5)$). При этом вершина $(3, -4)$ перейдет в точку $(3, 4)$.
2. Построим график уравнения $y - x + 1 = 0$.
Выразим $y$: $y = x - 1$. Это линейная функция, графиком которой является прямая.
Для построения прямой найдем координаты двух точек:
При $x = 1$, $y = 1 - 1 = 0$. Точка $(1, 0)$.
При $x = 6$, $y = 6 - 1 = 5$. Точка $(6, 5)$.
3. Построим оба графика в одной системе координат. Точки пересечения графиков являются решениями системы.
Графики пересекаются в трех точках. Из построения видно, что это точки с координатами $(1, 0)$, $(4, 3)$ и $(6, 5)$.
Ответ: $(1, 0)$, $(4, 3)$, $(6, 5)$.
б) Решим систему уравнений $\{_{xy + y = 4}^{x^2 + y^2 - 2y = 8}\}$ графически.
1. Проанализируем и построим график первого уравнения: $x^2 + y^2 - 2y = 8$.
Преобразуем уравнение, выделив полный квадрат относительно переменной $y$:
$x^2 + (y^2 - 2y + 1) - 1 = 8$
$x^2 + (y - 1)^2 = 9$
$x^2 + (y - 1)^2 = 3^2$
Это стандартное уравнение окружности с центром в точке $C(0, 1)$ и радиусом $R = 3$.
2. Проанализируем и построим график второго уравнения: $xy + y = 4$.
Вынесем $y$ за скобки: $y(x + 1) = 4$.
Если $x \neq -1$, то $y = \frac{4}{x+1}$.
Графиком этой функции является гипербола. Ее можно получить сдвигом графика функции $y = 4/x$ на 1 единицу влево вдоль оси Ox. Асимптоты гиперболы: вертикальная $x = -1$ и горизонтальная $y = 0$.
3. Построим окружность и гиперболу в одной системе координат. Координаты точек пересечения графиков будут решениями системы.
Из графиков видно, что они пересекаются в двух точках. Определим их координаты.
Первая точка пересечения имеет координаты $(3, 1)$.
Вторая точка пересечения имеет координаты $(0, 4)$.
Выполним проверку для найденных точек:
Для точки $(3, 1)$: $3^2 + (1-1)^2 = 9+0=9$ и $3 \cdot 1 + 1 = 4$. Оба равенства верны.
Для точки $(0, 4)$: $0^2 + (4-1)^2 = 0+9=9$ и $0 \cdot 4 + 4 = 4$. Оба равенства верны.
Таким образом, решениями системы являются две пары чисел.
Ответ: $(3, 1)$, $(0, 4)$.
Другие задания:
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