To solve the system of equations using the elimination method, we start with the given equations:
- \(-4x + 9y = -13\) (Equation 1)
- \(-x - 2y = 18\) (Equation 2)
First, we want to eliminate one of the variables. Let's eliminate \(x\). To do this, we can multiply Equation 2 by 4 so that the coefficients of \(x\) in both equations are the same:
\[ 4(-x - 2y) = 4(18) \] This simplifies to: \[ -4x - 8y = 72 \quad \text{(Equation 3)} \]
Now we have the modified Equation 3 and Equation 1:
- \(-4x + 9y = -13\) (Equation 1)
- \(-4x - 8y = 72\) (Equation 3)
Next, we can subtract Equation 3 from Equation 1 to eliminate \(x\):
\[ (-4x + 9y) - (-4x - 8y) = -13 - 72 \]
This simplifies to: \[ 9y + 8y = -13 - 72 \]
Combining like terms gives: \[ 17y = -85 \]
Now, we divide both sides by 17 to solve for \(y\): \[ y = -5 \]
Now that we have \(y\), we can substitute this value back into either original equation to solve for \(x\). We will use Equation 2:
\[ -x - 2(-5) = 18 \] This simplifies to: \[ -x + 10 = 18 \]
Subtracting 10 from both sides results in: \[ -x = 8 \]
Multiplying both sides by -1 gives us: \[ x = -8 \]
Thus, the solution to the system of equations is: \[ \boxed{(-8, -5)} \]