To solve the system of equations given by:
- \( y = -2x - 2 \) (Equation 1)
- \( 2x - 2y = 5 \) (Equation 2)
we can substitute Equation 1 into Equation 2 to eliminate \( y \).
Substituting \( y \) from Equation 1 into Equation 2:
\[ 2x - 2(-2x - 2) = 5 \]
Now simplify it:
\[ 2x + 4x + 4 = 5 \]
Combine like terms:
\[ 6x + 4 = 5 \]
Next, subtract 4 from both sides:
\[ 6x = 1 \]
Now, divide by 6:
\[ x = \frac{1}{6} \]
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):
\[ y = -2\left(\frac{1}{6}\right) - 2 \]
Calculating it gives us:
\[ y = -\frac{2}{6} - 2 = -\frac{1}{3} - 2 = -\frac{1}{3} - \frac{6}{3} = -\frac{7}{3} \]
So, the solution to the system of equations is:
\[ \left( x, y \right) = \left( \frac{1}{6}, -\frac{7}{3} \right) \]