To solve the system of equations
\[ -9x + 8y = 12 \quad (1) \]
\[ x - 8y = 20 \quad (2), \]
we can use either the substitution method or the elimination method. Here, we'll solve it using the elimination method.
First, let's align the two equations for elimination. We can multiply equation (2) by 9 to make the coefficients of \(x\) have the same magnitude:
\[ 9(x - 8y) = 9(20) \quad \Rightarrow \quad 9x - 72y = 180 \quad (3) \]
Now we have:
- \( -9x + 8y = 12\)
- \( 9x - 72y = 180\)
Next, we can add equations (1) and (3) together:
\[ (-9x + 8y) + (9x - 72y) = 12 + 180 \]
This simplifies to:
\[ (0) + (8y - 72y) = 192 \]
\[ -64y = 192 \]
Now, divide both sides by -64:
\[ y = \frac{192}{-64} = -3 \]
Now that we have \(y\), we can substitute this value back into one of the original equations to find \(x\). We can use equation (2):
\[ x - 8(-3) = 20 \]
\[ x + 24 = 20 \]
Subtracting 24 from both sides gives:
\[ x = 20 - 24 = -4 \]
Thus, the solution to the system of equations is:
\[ x = -4, \quad y = -3 \]
To summarize, the solution is \((x, y) = (-4, -3)\).