Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = x^2 + 2x - 6$
$2 = 2x - y$

Explanation:

Step1: Isolate y in linear equation

Rearrange $2=2x-y$ to solve for $y$.
$y=2x-2$

Step2: Substitute into quadratic equation

Replace $y$ in $y=x^2+2x-6$ with $2x-2$.
$2x-2=x^2+2x-6$

Step3: Simplify to quadratic equation

Cancel terms and rearrange to standard form.
$0=x^2-4$

Step4: Solve for x

Factor and solve the quadratic equation.
$x^2=4 \implies x=2 \text{ or } x=-2$

Step5: Find corresponding y-values

Substitute $x=2$ and $x=-2$ into $y=2x-2$.
For $x=2$: $y=2(2)-2=2$
For $x=-2$: $y=2(-2)-2=-6$

Answer:

$(2, 2)$ and $(-2, -6)$