To solve the system of equations using elimination, let's first express the equations:
- \( 3x - 4y = 40 \) (Equation 1)
- \( 5x - 5y = 55 \) (Equation 2)
We want to eliminate one of the variables. We can do this by manipulating the equations.
Let's multiply Equation 1 by 5 and Equation 2 by 3 to align the coefficients of \(x\):
\[ 5(3x - 4y) = 5(40) \implies 15x - 20y = 200 \quad \text{(Equation 3)} \] \[ 3(5x - 5y) = 3(55) \implies 15x - 15y = 165 \quad \text{(Equation 4)} \]
Now we can subtract Equation 4 from Equation 3:
\[ (15x - 20y) - (15x - 15y) = 200 - 165 \] \[ -20y + 15y = 35 \] \[ -5y = 35 \] \[ y = -7 \]
Now that we found \(y\), we can substitute it back into one of the original equations to find \(x\). We'll use Equation 1:
\[ 3x - 4(-7) = 40 \] \[ 3x + 28 = 40 \] \[ 3x = 40 - 28 \] \[ 3x = 12 \] \[ x = 4 \]
So the solution to the system of equations is \((x, y) = (4, -7)\).
The final answer is: \[ \boxed{(4, -7)} \]