To solve the system of equations given by:
- \( 3x - 4y = 40 \)
- \( 5x - 5y = 55 \)
we can use the method of substitution or elimination. Here, I'll use the elimination method.
First, we can simplify the second equation:
\[ 5x - 5y = 55 \implies x - y = 11 \quad \text{(divide everything by 5)} \]
Now we have:
- \( 3x - 4y = 40 \)
- \( x - y = 11 \)
From the second equation, we can express \( x \) in terms of \( y \):
\[ x = y + 11 \]
Now substitute this expression for \( x \) into the first equation:
\[ 3(y + 11) - 4y = 40 \]
Distributing \( 3 \):
\[ 3y + 33 - 4y = 40 \]
Now combine like terms:
\[ -1y + 33 = 40 \]
Subtract 33 from both sides:
\[ -y = 7 \]
Now, multiply both sides by -1:
\[ y = -7 \]
Now that we have \( y \), we substitute back to find \( x \):
\[ x = y + 11 = -7 + 11 = 4 \]
Thus, the solution to the system of equations is:
\[ \boxed{(4, -7)} \]
To verify, we can substitute these values back into the original equations:
- \( 3(4) - 4(-7) = 12 + 28 = 40 \) (True)
- \( 5(4) - 5(-7) = 20 + 35 = 55 \) (True)
Since both equations hold true, our solution \( (4, -7) \) is confirmed correct.