To solve the system of equations, we can set the two expressions for \( y \) equal to each other because they both equal \( y \).
The two equations are:
- \( y = \frac{1}{2}x - 4 \) (Equation 1)
- \( y = -2x - 9 \) (Equation 2)
Now we'll set them equal to each other:
\[ \frac{1}{2}x - 4 = -2x - 9 \]
To eliminate the fraction, we can multiply every term by 2:
\[ x - 8 = -4x - 18 \]
Next, we'll add \( 4x \) to both sides:
\[ x + 4x - 8 = -18 \]
This simplifies to:
\[ 5x - 8 = -18 \]
Now, add 8 to both sides:
\[ 5x = -10 \]
Divide by 5:
\[ x = -2 \]
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We'll use Equation 1:
\[ y = \frac{1}{2}(-2) - 4 \]
\[ y = -1 - 4 \]
\[ y = -5 \]
Thus, the solution to the system of equations is \( (x, y) = (-2, -5) \).
Verifying with Equation 2:
\[ y = -2(-2) - 9 \]
\[ y = 4 - 9 \]
\[ y = -5 \]
Both equations confirm that the solution is valid.
The answer is \( (-2, -5) \).