Пример №38 из задания 20

На ленте по разные стороны от середины отмечены две тонкие поперечные полоски: синяя и красная. Если разрезать ленту по красной полоске, то одна часть будет на \(35\) см длиннее другой. Если разрезать ленту по синей полоске, то одна часть будет на \(75\) см длиннее другой. Найдите расстояние (в сантиметрах) между красной и синей полосками.


Решение

Пусть \(x\) — расстояние от начала ленты до синей полоски; \(y\) — расстояние между синей и красной полосками, тогда \(z\) — расстояние от красной полоски до конца ленты.

Если разрезать ленту по красной полоске, то одна часть будет на \(35\) см длиннее другой, значит \(x+y=z+35\).

Если разрезать ту же самую ленту по синей полоске, то одна часть будет на \(75\) см длиннее другой, значит можно составить уравнение \(y+z=x+75\).

Получилась система уравнений:

\(\begin{cases} {x+y=z+35} \\ {y+z=x+75} \end{cases}\);

\(\begin{cases} {x+y-z=35} \\ {-x+y+z=75} \end{cases}\).

Сложим уравнения: \(x-x+y+y-z+z=35+75\);

\(2y=110\);

\(y=55\).

Получилось, что расстояние между красной и синей полосками равно \(55\) см.

Ответ: \(55\).


Источник: ЕГЭ 2021. Математика. Базовый уровень. 50 вариантов. Типовые варианты экзаменационных заданий. (вариант №12) (Купить книгу)

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