To solve the system of equations:
- \( 3x - 4y = 40 \) (Equation 1)
- \( 5x - 5y = 55 \) (Equation 2)
First, we can simplify Equation 2. Dividing every term by 5 gives us:
\[ x - y = 11 \tag{Equation 2 simplified} \]
Next, we can express \( x \) in terms of \( y \) from Equation 2:
\[ x = y + 11 \tag{Equation 3} \]
Now, we can substitute Equation 3 into Equation 1:
\[ 3(y + 11) - 4y = 40 \]
Expanding this gives:
\[ 3y + 33 - 4y = 40 \]
Combining like terms:
\[ -y + 33 = 40 \]
To isolate \( y \), we can subtract 33 from both sides:
\[ -y = 40 - 33 \]
\[ -y = 7 \]
Multiplying by -1 gives:
\[ y = -7 \]
Now, we can substitute \( y = -7 \) back into Equation 3 to find \( x \):
\[ x = -7 + 11 \]
\[ x = 4 \]
Now we have the solution:
\[ x = 4, \quad y = -7 \]
To verify, we can substitute these values into the original equations:
-
For Equation 1: \[ 3(4) - 4(-7) = 12 + 28 = 40 \quad \text{(True)} \]
-
For Equation 2: \[ 5(4) - 5(-7) = 20 + 35 = 55 \quad \text{(True)} \]
Thus, the solution to the system of equations is:
\[ \boxed{(4, -7)} \]