To solve the system of equations:
- \( 6x - 7y = 37 \) (Equation 1)
- \( x + 14y = -9 \) (Equation 2)
We can use the substitution or elimination method. Here, I will use substitution.
First, let's solve Equation 2 for \( x \):
\[ x = -9 - 14y \]
Now, substitute this expression for \( x \) into Equation 1:
\[ 6(-9 - 14y) - 7y = 37 \]
Distributing the 6:
\[ -54 - 84y - 7y = 37 \]
Combine like terms:
\[ -54 - 91y = 37 \]
Now, isolate \( y \):
\[ -91y = 37 + 54 \] \[ -91y = 91 \] \[ y = \frac{91}{-91} = -1 \]
Now that we have \( y = -1 \), we can substitute back into the expression for \( x \):
\[ x = -9 - 14(-1) \] \[ x = -9 + 14 \] \[ x = 5 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (5, -1) \]
To verify, you can substitute \( x = 5 \) and \( y = -1 \) back into the original equations:
-
For \( 6x - 7y = 37 \): \[ 6(5) - 7(-1) = 30 + 7 = 37 \quad \text{(True)} \]
-
For \( x + 14y = -9 \): \[ 5 + 14(-1) = 5 - 14 = -9 \quad \text{(True)} \]
The solution satisfies both equations, confirming that the solution is correct.
Final answer: \[ (x, y) = (5, -1) \]