Chapter 11.6, Problem 71AYU

The difference of two numbers is 2 and the sum of their squares is 10. Find the numbers.

To determine

To find: The numbers $x$ and $y$

Answer to Problem 71AYU

Solution:

$\left(- 1, - 3\right)$ and $\left(3,1\right)$

Explanation of Solution

Given:

The difference of the two is 2 and the sum of their squares is 10.

$x - y = 2$    -----(Eq 1)

$x^2 + y^2 = 10$    -----(Eq 2)

Calculation:

Substituting $y = x - 2$ in Eq 2 we get

$x^2 + {\left(x - 2\right)}^2 = 10$

The above equation can be rewritten as

$2x^2 - 4x - 6 = 0$

$= >x^2 - 2x^2 - 3 = 0$

This can be factored as $\left(x + 1\right)\left(x - 3\right) = 0$

i.e $x = \text{-}1,3$

Substituting $x$ in Eq 2, we get $y = - 3,1$

The solutions to the equations are $\left(- 1, - 3\right)$ and $\left(3,1\right)$.