To solve the system of equations:
- \(-3x - 4y = -23\) (Equation 1)
- \(3x - 5y = 5\) (Equation 2)
we can use the elimination or substitution method. Here, I'll use the elimination method.
First, let's eliminate \(x\) by adding the two equations together. We'll multiply Equation 2 by 1 (to make it easier) and then add them:
\[ -3x - 4y = -23 \quad \text{(1)} \] \[ 3x - 5y = 5 \quad \text{(2)} \]
Add Equation 1 and Equation 2: \[ (-3x + 3x) + (-4y - 5y) = -23 + 5 \] This simplifies to: \[ 0 - 9y = -18 \] So: \[ -9y = -18 \] Dividing both sides by \(-9\): \[ y = 2 \]
Now that we have \(y\), we can find \(x\) by substituting \(y = 2\) back into one of the original equations. Using Equation 2:
\[ 3x - 5(2) = 5 \] This simplifies to: \[ 3x - 10 = 5 \] Adding 10 to both sides: \[ 3x = 15 \] Dividing both sides by 3: \[ x = 5 \]
So the solution to the system of equations is: \[ x = 5, \quad y = 2 \]
Therefore, the solution is \((x, y) = (5, 2)\).