To find the solution(s) of the graphed system of equations [tex]\(y = x^2 + x - 2\)[/tex] and [tex]\(y = 2x - 2\)[/tex], you need to determine the points where these two curves intersect.
Here is a step-by-step solution:
1. Set the equations equal to each other:
[tex]\[
x^2 + x - 2 = 2x - 2
\][/tex]
2. Rearrange the equation to set it to zero:
[tex]\[
x^2 + x - 2 - 2x + 2 = 0
\][/tex]
Simplify:
[tex]\[
x^2 - x = 0
\][/tex]
3. Factor the quadratic equation:
[tex]\[
x(x - 1) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
This gives us two solutions:
[tex]\[
x = 0 \quad \text{or} \quad x = 1
\][/tex]
5. Substitute these [tex]\(x\)[/tex]-values into one of the original equations to find the corresponding [tex]\(y\)[/tex]-values.
For [tex]\(x = 0\)[/tex]:
[tex]\[
y = 2(0) - 2 = -2
\][/tex]
For [tex]\(x = 1\)[/tex]:
[tex]\[
y = 2(1) - 2 = 0
\][/tex]
Thus, the points of intersection are [tex]\((0, -2)\)[/tex] and [tex]\((1, 0)\)[/tex]. These are the solutions of the graphed system of equations.
Therefore, the solution is:
[tex]\[
(0, -2) \text{ and } (1, 0)
\][/tex]
