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Номер 83, страница 30 - 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 83, страница 30.
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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 } 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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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{#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} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "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" => 1276438 "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" => "1107780" "type" => "task" ] "text" => "<p><strong>83.</strong> Решите графически систему уравнений:</p><p><strong>а)</strong></p><p>$\begin{cases} x + y = 3, \\ x^2 + y^2 = 9; \end{cases}$</p><p><strong>б)</strong></p><p>$\begin{cases} y - x = 1, \\ y = -(x - 1)^2 + 4; \end{cases}$</p><p><strong>в)</strong></p><p>$\begin{cases} y = (x - 1)^2 + 2, \\ xy = 2; \end{cases}$</p><p><strong>г)</strong></p><p>$\begin{cases} x^2 + y^2 = 25, \\ xy = 12. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "83-1.jpg" "alt" => null "width" => "1085" "height" => 452 "path" => "/media/algebra_09/soltan-u19/0-1/83-1.webp?ts=1752834155" ] ] ] #original: array:7 [ "id" => 1276438 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1107780" "type" => "task" ] "text" => "<p><strong>83.</strong> Решите графически систему уравнений:</p><p><strong>а)</strong></p><p>$\begin{cases} x + y = 3, \\ x^2 + y^2 = 9; \end{cases}$</p><p><strong>б)</strong></p><p>$\begin{cases} y - x = 1, \\ y = -(x - 1)^2 + 4; \end{cases}$</p><p><strong>в)</strong></p><p>$\begin{cases} y = (x - 1)^2 + 2, \\ xy = 2; \end{cases}$</p><p><strong>г)</strong></p><p>$\begin{cases} x^2 + y^2 = 25, \\ xy = 12. \end{cases}$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "83-1.jpg" "alt" => null "width" => "1085" "height" => 452 "path" => "/media/algebra_09/soltan-u19/0-1/83-1.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" => 1277656 "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" => "1107780" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "83-1.jpg" "alt" => null "width" => "1213" "height" => 2424 "path" => "/media/algebra_09/soltan-u19/1-1/83-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "83-2.jpg" "alt" => null "width" => "1258" "height" => 1188 "path" => "/media/algebra_09/soltan-u19/1-1/83-2.webp?ts=1752830799" ] ] ] #original: array:6 [ "id" => 1277656 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1107780" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "83-1.jpg" "alt" => null "width" => "1213" "height" => 2424 "path" => "/media/algebra_09/soltan-u19/1-1/83-1.webp?ts=1752830799" ] 1 => array:5 [ "name" => "83-2.jpg" "alt" => null "width" => "1258" "height" => 1188 "path" => "/media/algebra_09/soltan-u19/1-1/83-2.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" => 1553684 "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" => "1107780" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для решения системы уравнений $\begin{cases} x + y = 3 \\ x^2 + y^2 = 9 \end{cases}$ графически, построим графики каждого уравнения на одной координатной плоскости.</p><p>Первое уравнение, $x + y = 3$, можно переписать в виде $y = 3 - x$. Это уравнение прямой линии. Для её построения достаточно двух точек. Например, если $x = 0$, то $y = 3$ (точка $(0, 3)$), и если $y = 0$, то $x = 3$ (точка $(3, 0)$).</p><p>Второе уравнение, $x^2 + y^2 = 9$, является уравнением окружности с центром в начале координат $(0, 0)$ и радиусом $r = \sqrt{9} = 3$.</p><p>Теперь построим оба графика. Прямая проходит через точки $(0, 3)$ и $(3, 0)$. Окружность с центром в $(0, 0)$ и радиусом 3 также проходит через эти две точки. Точки пересечения графиков и являются решениями системы. В данном случае это точки $(0, 3)$ и $(3, 0)$.</p><p><strong>Ответ:</strong> $(0, 3), (3, 0)$.</p><p><strong>б)</strong></p><p>Рассмотрим систему $\begin{cases} y - x = 1 \\ y = -(x - 1)^2 + 4 \end{cases}$.</p><p>Первое уравнение, $y - x = 1$, преобразуется к виду $y = x + 1$. Это уравнение прямой линии. Построим её по двум точкам: если $x = 0$, то $y = 1$ (точка $(0, 1)$), и если $x = 2$, то $y = 3$ (точка $(2, 3)$).</p><p>Второе уравнение, $y = -(x - 1)^2 + 4$, является уравнением параболы. Ветви параболы направлены вниз, так как коэффициент при скобке в квадрате отрицателен. Вершина параболы находится в точке $(1, 4)$.</p><p>Найдём точки пересечения графиков. Из графиков видно, что прямая и парабола пересекаются в двух точках. Графически можно определить, что одна из точек пересечения — $(-1, 0)$, так как $y(-1) = -1 + 1 = 0$ для прямой и $y(-1) = -(-1 - 1)^2 + 4 = -(-2)^2 + 4 = -4 + 4 = 0$ для параболы. Другая точка пересечения — $(2, 3)$, так как $y(2) = 2 + 1 = 3$ для прямой и $y(2) = -(2 - 1)^2 + 4 = -1 + 4 = 3$ для параболы.</p><p>Следовательно, точки пересечения: $(-1, 0)$ и $(2, 3)$.</p><p><strong>Ответ:</strong> $(-1, 0), (2, 3)$.</p><p><strong>в)</strong></p><p>Рассмотрим систему $\begin{cases} y = (x - 1)^2 + 2 \\ xy = 2 \end{cases}$.</p><p>Первое уравнение, $y = (x - 1)^2 + 2$, задаёт параболу с ветвями, направленными вверх, и вершиной в точке $(1, 2)$. Минимальное значение $y$ для этой параболы равно 2.</p><p>Второе уравнение, $xy = 2$, можно записать как $y = 2/x$. Это уравнение гиперболы, ветви которой расположены в первом и третьем координатных квадрантах.</p><p>Построим графики. Парабола имеет вершину в точке $(1, 2)$ и проходит через точки $(0, 3)$ и $(2, 3)$. Гипербола проходит через точки $(1, 2)$, $(2, 1)$, $(-1, -2)$ и $(-2, -1)$.</p><p>Из графиков видно, что они пересекаются в одной точке — вершине параболы $(1, 2)$, которая также лежит на гиперболе (так как $1 \cdot 2 = 2$). Поскольку все точки параболы имеют ординату $y \ge 2$, а точки гиперболы в первой четверти при $x > 1$ имеют ординату $y < 2$, других пересечений в первой четверти нет. В третьей четверти пересечений нет, так как парабола целиком лежит выше оси абсцисс. Таким образом, система имеет единственное решение.</p><p><strong>Ответ:</strong> $(1, 2)$.</p><p><strong>г)</strong></p><p>Рассмотрим систему $\begin{cases} x^2 + y^2 = 25 \\ xy = 12 \end{cases}$.</p><p>Первое уравнение, $x^2 + y^2 = 25$, является уравнением окружности с центром в начале координат $(0, 0)$ и радиусом $r = \sqrt{25} = 5$.</p><p>Второе уравнение, $xy = 12$, или $y = 12/x$, является уравнением гиперболы с ветвями в первом и третьем квадрантах.</p><p>Построим графики на одной координатной плоскости. Окружность радиуса 5 проходит через точки $(5, 0)$, $(0, 5)$, $(-5, 0)$, $(0, -5)$. Гипербола проходит, например, через точки $(3, 4)$, $(4, 3)$, $(-3, -4)$ и $(-4, -3)$.</p><p>Заметим, что точки $(3, 4)$ и $(4, 3)$ лежат на гиперболе. Проверим, лежат ли они на окружности: $3^2 + 4^2 = 9 + 16 = 25$. Да, лежат. Аналогично, точки $(-3, -4)$ и $(-4, -3)$ лежат на гиперболе. Проверим их для окружности: $(-3)^2 + (-4)^2 = 9 + 16 = 25$. Они также лежат на окружности. Графики пересекаются в четырех точках.</p><p><strong>Ответ:</strong> $(3, 4), (4, 3), (-3, -4), (-4, -3)$.</p>" ] #original: array:6 [ "id" => 1553684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1107780" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для решения системы уравнений $\begin{cases} x + y = 3 \\ x^2 + y^2 = 9 \end{cases}$ графически, построим графики каждого уравнения на одной координатной плоскости.</p><p>Первое уравнение, $x + y = 3$, можно переписать в виде $y = 3 - x$. Это уравнение прямой линии. Для её построения достаточно двух точек. Например, если $x = 0$, то $y = 3$ (точка $(0, 3)$), и если $y = 0$, то $x = 3$ (точка $(3, 0)$).</p><p>Второе уравнение, $x^2 + y^2 = 9$, является уравнением окружности с центром в начале координат $(0, 0)$ и радиусом $r = \sqrt{9} = 3$.</p><p>Теперь построим оба графика. Прямая проходит через точки $(0, 3)$ и $(3, 0)$. Окружность с центром в $(0, 0)$ и радиусом 3 также проходит через эти две точки. Точки пересечения графиков и являются решениями системы. В данном случае это точки $(0, 3)$ и $(3, 0)$.</p><p><strong>Ответ:</strong> $(0, 3), (3, 0)$.</p><p><strong>б)</strong></p><p>Рассмотрим систему $\begin{cases} y - x = 1 \\ y = -(x - 1)^2 + 4 \end{cases}$.</p><p>Первое уравнение, $y - x = 1$, преобразуется к виду $y = x + 1$. Это уравнение прямой линии. Построим её по двум точкам: если $x = 0$, то $y = 1$ (точка $(0, 1)$), и если $x = 2$, то $y = 3$ (точка $(2, 3)$).</p><p>Второе уравнение, $y = -(x - 1)^2 + 4$, является уравнением параболы. Ветви параболы направлены вниз, так как коэффициент при скобке в квадрате отрицателен. Вершина параболы находится в точке $(1, 4)$.</p><p>Найдём точки пересечения графиков. Из графиков видно, что прямая и парабола пересекаются в двух точках. Графически можно определить, что одна из точек пересечения — $(-1, 0)$, так как $y(-1) = -1 + 1 = 0$ для прямой и $y(-1) = -(-1 - 1)^2 + 4 = -(-2)^2 + 4 = -4 + 4 = 0$ для параболы. Другая точка пересечения — $(2, 3)$, так как $y(2) = 2 + 1 = 3$ для прямой и $y(2) = -(2 - 1)^2 + 4 = -1 + 4 = 3$ для параболы.</p><p>Следовательно, точки пересечения: $(-1, 0)$ и $(2, 3)$.</p><p><strong>Ответ:</strong> $(-1, 0), (2, 3)$.</p><p><strong>в)</strong></p><p>Рассмотрим систему $\begin{cases} y = (x - 1)^2 + 2 \\ xy = 2 \end{cases}$.</p><p>Первое уравнение, $y = (x - 1)^2 + 2$, задаёт параболу с ветвями, направленными вверх, и вершиной в точке $(1, 2)$. Минимальное значение $y$ для этой параболы равно 2.</p><p>Второе уравнение, $xy = 2$, можно записать как $y = 2/x$. Это уравнение гиперболы, ветви которой расположены в первом и третьем координатных квадрантах.</p><p>Построим графики. Парабола имеет вершину в точке $(1, 2)$ и проходит через точки $(0, 3)$ и $(2, 3)$. Гипербола проходит через точки $(1, 2)$, $(2, 1)$, $(-1, -2)$ и $(-2, -1)$.</p><p>Из графиков видно, что они пересекаются в одной точке — вершине параболы $(1, 2)$, которая также лежит на гиперболе (так как $1 \cdot 2 = 2$). Поскольку все точки параболы имеют ординату $y \ge 2$, а точки гиперболы в первой четверти при $x > 1$ имеют ординату $y < 2$, других пересечений в первой четверти нет. В третьей четверти пересечений нет, так как парабола целиком лежит выше оси абсцисс. Таким образом, система имеет единственное решение.</p><p><strong>Ответ:</strong> $(1, 2)$.</p><p><strong>г)</strong></p><p>Рассмотрим систему $\begin{cases} x^2 + y^2 = 25 \\ xy = 12 \end{cases}$.</p><p>Первое уравнение, $x^2 + y^2 = 25$, является уравнением окружности с центром в начале координат $(0, 0)$ и радиусом $r = \sqrt{25} = 5$.</p><p>Второе уравнение, $xy = 12$, или $y = 12/x$, является уравнением гиперболы с ветвями в первом и третьем квадрантах.</p><p>Построим графики на одной координатной плоскости. Окружность радиуса 5 проходит через точки $(5, 0)$, $(0, 5)$, $(-5, 0)$, $(0, -5)$. Гипербола проходит, например, через точки $(3, 4)$, $(4, 3)$, $(-3, -4)$ и $(-4, -3)$.</p><p>Заметим, что точки $(3, 4)$ и $(4, 3)$ лежат на гиперболе. Проверим, лежат ли они на окружности: $3^2 + 4^2 = 9 + 16 = 25$. Да, лежат. Аналогично, точки $(-3, -4)$ и $(-4, -3)$ лежат на гиперболе. Проверим их для окружности: $(-3)^2 + (-4)^2 = 9 + 16 = 25$. Они также лежат на окружности. Графики пересекаются в четырех точках.</p><p><strong>Ответ:</strong> $(3, 4), (4, 3), (-3, -4), (-4, -3)$.</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" => "1107781" "type" => "task" ] "previous" => array:2 [ "refs" => "1107779" "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: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:3 [ 0 => App\Models\Term {#1132 #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 {#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: [] 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"field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Term {#1137 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "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 {#1138 #items: array:1 [ 0 => App\Models\Term {#1139 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false 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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 {#1125} "field_class" => Illuminate\Database\Eloquent\Collection {#1127} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129} "field_author" => Illuminate\Database\Eloquent\Collection {#1131} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_country" => Illuminate\Database\Eloquent\Collection {#1138} "field_city" => Illuminate\Database\Eloquent\Collection {#1140} "field_series" => Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => 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 {#1166 #items: array:1 [ 0 => App\Models\BookPage {#1165 #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" => 1097772 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-30" "field_display_title" => "30" "field_folder" => "folder1" "field_image_name" => "30" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1168 #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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1196 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Element {#1197 #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" => 1260923 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1260923 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 …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" => "1097773" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097771" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2006 #items: array:6 [ 0 => App\Models\Task {#1206 #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" => 1107775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-30" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1209 …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 {#1207 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1312 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-30" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1209 …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 {#1207 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1312 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …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 {#1366 #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" => 1107776 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-79" "field_display_title" => "79" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1367 …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 {#1369 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1370 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1403 …2} "content" => Illuminate\Database\Eloquent\Collection {#1440 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1465 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107776 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-79" "field_display_title" => "79" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1367 …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 {#1369 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1370 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1403 …2} "content" => Illuminate\Database\Eloquent\Collection {#1440 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1465 …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 {#1494 #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" => 1107777 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-80" "field_display_title" => "80" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1495 …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 {#1497 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1498 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1531 …2} "content" => Illuminate\Database\Eloquent\Collection {#1568 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1593 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1107777 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-80" "field_display_title" => "80" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1495 …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 {#1497 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1498 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1531 …2} "content" => Illuminate\Database\Eloquent\Collection {#1568 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1593 …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 {#1622 #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" => 1107778 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "30" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-81" "field_display_title" => "81" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1623 …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 {#1625 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1626 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1659 …2} "content" => Illuminate\Database\Eloquent\Collection {#1696 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1721 …2} "page" => array:2 [ …2] ] 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№83 (с. 30)
Условие. №83 (с. 30)
скриншот условия
83. Решите графически систему уравнений:
а)
$\begin{cases} x + y = 3, \\ x^2 + y^2 = 9; \end{cases}$
б)
$\begin{cases} y - x = 1, \\ y = -(x - 1)^2 + 4; \end{cases}$
в)
$\begin{cases} y = (x - 1)^2 + 2, \\ xy = 2; \end{cases}$
г)
$\begin{cases} x^2 + y^2 = 25, \\ xy = 12. \end{cases}$
Решение. №83 (с. 30)
Решение 2 (rus). №83 (с. 30)
а)
Для решения системы уравнений $\begin{cases} x + y = 3 \\ x^2 + y^2 = 9 \end{cases}$ графически, построим графики каждого уравнения на одной координатной плоскости.
Первое уравнение, $x + y = 3$, можно переписать в виде $y = 3 - x$. Это уравнение прямой линии. Для её построения достаточно двух точек. Например, если $x = 0$, то $y = 3$ (точка $(0, 3)$), и если $y = 0$, то $x = 3$ (точка $(3, 0)$).
Второе уравнение, $x^2 + y^2 = 9$, является уравнением окружности с центром в начале координат $(0, 0)$ и радиусом $r = \sqrt{9} = 3$.
Теперь построим оба графика. Прямая проходит через точки $(0, 3)$ и $(3, 0)$. Окружность с центром в $(0, 0)$ и радиусом 3 также проходит через эти две точки. Точки пересечения графиков и являются решениями системы. В данном случае это точки $(0, 3)$ и $(3, 0)$.
Ответ: $(0, 3), (3, 0)$.
б)
Рассмотрим систему $\begin{cases} y - x = 1 \\ y = -(x - 1)^2 + 4 \end{cases}$.
Первое уравнение, $y - x = 1$, преобразуется к виду $y = x + 1$. Это уравнение прямой линии. Построим её по двум точкам: если $x = 0$, то $y = 1$ (точка $(0, 1)$), и если $x = 2$, то $y = 3$ (точка $(2, 3)$).
Второе уравнение, $y = -(x - 1)^2 + 4$, является уравнением параболы. Ветви параболы направлены вниз, так как коэффициент при скобке в квадрате отрицателен. Вершина параболы находится в точке $(1, 4)$.
Найдём точки пересечения графиков. Из графиков видно, что прямая и парабола пересекаются в двух точках. Графически можно определить, что одна из точек пересечения — $(-1, 0)$, так как $y(-1) = -1 + 1 = 0$ для прямой и $y(-1) = -(-1 - 1)^2 + 4 = -(-2)^2 + 4 = -4 + 4 = 0$ для параболы. Другая точка пересечения — $(2, 3)$, так как $y(2) = 2 + 1 = 3$ для прямой и $y(2) = -(2 - 1)^2 + 4 = -1 + 4 = 3$ для параболы.
Следовательно, точки пересечения: $(-1, 0)$ и $(2, 3)$.
Ответ: $(-1, 0), (2, 3)$.
в)
Рассмотрим систему $\begin{cases} y = (x - 1)^2 + 2 \\ xy = 2 \end{cases}$.
Первое уравнение, $y = (x - 1)^2 + 2$, задаёт параболу с ветвями, направленными вверх, и вершиной в точке $(1, 2)$. Минимальное значение $y$ для этой параболы равно 2.
Второе уравнение, $xy = 2$, можно записать как $y = 2/x$. Это уравнение гиперболы, ветви которой расположены в первом и третьем координатных квадрантах.
Построим графики. Парабола имеет вершину в точке $(1, 2)$ и проходит через точки $(0, 3)$ и $(2, 3)$. Гипербола проходит через точки $(1, 2)$, $(2, 1)$, $(-1, -2)$ и $(-2, -1)$.
Из графиков видно, что они пересекаются в одной точке — вершине параболы $(1, 2)$, которая также лежит на гиперболе (так как $1 \cdot 2 = 2$). Поскольку все точки параболы имеют ординату $y \ge 2$, а точки гиперболы в первой четверти при $x > 1$ имеют ординату $y < 2$, других пересечений в первой четверти нет. В третьей четверти пересечений нет, так как парабола целиком лежит выше оси абсцисс. Таким образом, система имеет единственное решение.
Ответ: $(1, 2)$.
г)
Рассмотрим систему $\begin{cases} x^2 + y^2 = 25 \\ xy = 12 \end{cases}$.
Первое уравнение, $x^2 + y^2 = 25$, является уравнением окружности с центром в начале координат $(0, 0)$ и радиусом $r = \sqrt{25} = 5$.
Второе уравнение, $xy = 12$, или $y = 12/x$, является уравнением гиперболы с ветвями в первом и третьем квадрантах.
Построим графики на одной координатной плоскости. Окружность радиуса 5 проходит через точки $(5, 0)$, $(0, 5)$, $(-5, 0)$, $(0, -5)$. Гипербола проходит, например, через точки $(3, 4)$, $(4, 3)$, $(-3, -4)$ и $(-4, -3)$.
Заметим, что точки $(3, 4)$ и $(4, 3)$ лежат на гиперболе. Проверим, лежат ли они на окружности: $3^2 + 4^2 = 9 + 16 = 25$. Да, лежат. Аналогично, точки $(-3, -4)$ и $(-4, -3)$ лежат на гиперболе. Проверим их для окружности: $(-3)^2 + (-4)^2 = 9 + 16 = 25$. Они также лежат на окружности. Графики пересекаются в четырех точках.
Ответ: $(3, 4), (4, 3), (-3, -4), (-4, -3)$.
Другие задания:
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