# How do you find two positive numbers such that the sum of the squares of the two numbers is 169 and the difference between the two numbers is 7?

Dec 11, 2016

5, 12

#### Explanation:

Let $x$ be the larger of the 2 positive numbers
Let $y$ be the smaller of the 2 positive numbers

${x}^{2} + {y}^{2} = 169$

$x - y = 7$

$\implies x = 7 + y$

${x}^{2} + {y}^{2} = 169$

$\implies {\left(y + 7\right)}^{2} + {y}^{2} = 169$

$\implies {y}^{2} + 14 y + 49 + {y}^{2} = 169$

$\implies 2 {y}^{2} + 14 y - 120 = 0$

$\implies 2 \left({y}^{2} + 7 y - 60\right) = 0$

$\implies {y}^{2} + 7 y - 60 = 0$

$\implies \left(y + 12\right) \left(y - 5\right) = 0$

$\implies y = - 12 , y = 5$

However, we want $y > 0$

$\implies y = 5$

$x - y = 7$

$\implies x - 5 = 7$

$\implies x = 12$