Question

find the solution set for the given system by graphing both of the systems equations in the same rectangular coordinate system and finding all points of intersection.
$(x - 5)^2+(y + 2)^2=25$
$y=x - 2$

Explanation:

Step1: Substitute \(y = x - 2\) into the circle - equation

Substitute \(y\) in \((x - 5)^2+(y + 2)^2=25\). We get \((x - 5)^2+(x-2 + 2)^2=25\).
Simplify it: \((x - 5)^2+x^2=25\).
Expand \((x - 5)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\), so \(x^2-10x + 25+x^2=25\).
Combine like - terms: \(2x^2-10x=0\).
Factor out \(2x\): \(2x(x - 5)=0\).

Step2: Solve for \(x\)

Set each factor equal to zero:
If \(2x=0\), then \(x = 0\).
If \(x - 5=0\), then \(x = 5\).

Step3: Solve for \(y\)

When \(x = 0\), substitute \(x\) into \(y=x - 2\), so \(y=0 - 2=-2\).
When \(x = 5\), substitute \(x\) into \(y=x - 2\), so \(y=5 - 2=3\).

Answer:

The solution set is \(\{(0,-2),(5,3)\}\)