CEMC Fermat

Algebra Difficulty 2.3 Junior Find the answer

If $10x+y=75$ and $10y+x=57$ for some positive integers $x$ and $y$ , what is the value of $x+y$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $10x+y=75$ and $10y+x=57$ , then $(10x+y)+(10y+x)=75+57$ and so $11x+11y=132$ . Dividing by 11, we get $x+y=12$ . (We could have noticed initially that $(x, y)=(7,5)$ is a pair that satisfies the two equations, thence concluding that $x+y=12$ .)

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