# Question #b1d06

Apr 22, 2017

Substitute the right side of the second equation for y into the first equation.
Solve for the x coordinates
Use the second equation to find the corresponding y coordinates.

#### Explanation:

Given:

${x}^{2} + {y}^{2} = 25 \text{ [1]}$
$y = x - 7 \text{ [2]}$

Substitute $x - 7$ for y into equation [1]:

${x}^{2} + {\left(x - 7\right)}^{2} = 25$

Expand the square:

${x}^{2} + {x}^{2} - 14 x + 49 = 25$

Combine like terms:

$2 {x}^{2} - 14 x + 24 = 0$

Divide both sides by 2:

${x}^{2} - 7 x + 12 = 0$

Factor:

$\left(x - 3\right) \left(x - 4\right) = 0$

$x = 3$ and $x = 4$

Use equation [2] to find the y coordinates:

$y = x - 7$

$y = 3 - 7$ and $y = 4 - 7$

$y = - 4$ and $y = - 3$

The points that solve the two equation are $\left(3 , - 4\right)$ and $\left(4 , - 3\right)$

Here is a graph of the two equations:

graph{(x^2+y^2-25)(y-x+7)=0 [-10, 10, -5, 5]}