To solve the system of equations:
[tex]\[
\begin{cases}
y = 3x - 3 \\
y = x^2 + 5x - 2
\end{cases}
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. The strategy involves setting the equations equal to each other since both are expressions for [tex]\(y\)[/tex].
1. Set the right-hand sides of the equations equal to each other:
[tex]\[
3x - 3 = x^2 + 5x - 2
\][/tex]
2. Rearrange all the terms to one side to set the equation to zero:
[tex]\[
3x - 3 - (x^2 + 5x - 2) = 0
\][/tex]
[tex]\[
3x - 3 - x^2 - 5x + 2 = 0
\][/tex]
3. Combine like terms:
[tex]\[
-x^2 - 2x - 1 = 0
\][/tex]
4. Multiply through by -1 to simplify the quadratic equation:
[tex]\[
x^2 + 2x + 1 = 0
\][/tex]
5. Notice that this quadratic can be factored as a perfect square:
[tex]\[
(x + 1)^2 = 0
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 1 = 0
\][/tex]
[tex]\[
x = -1
\][/tex]
7. Substitute [tex]\(x = -1\)[/tex] back into the first equation to find [tex]\(y\)[/tex]:
[tex]\[
y = 3(-1) - 3
\][/tex]
[tex]\[
y = -3 - 3
\][/tex]
[tex]\[
y = -6
\][/tex]
8. The solution to the system of equations is the point where both curves intersect:
[tex]\[
(x, y) = (-1, -6)
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{(-1, -6)}
\][/tex]
