Answer :

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]