# 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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