To solve the system of equations using the linear combination method, we need to eliminate one of the variables. The given equations are:
- \( 3x + 7y = 31 \) (Equation 1)
- \( -3x - 2y = -1 \) (Equation 2)
First, we can multiply Equation 2 by -1 to make the coefficients of \( x \) opposites:
\[ 3x + 2y = 1 \quad \text{(Equation 3)} \]
Now, we have:
- \( 3x + 7y = 31 \)
- \( 3x + 2y = 1 \)
Next, we can subtract Equation 3 from Equation 1 to eliminate \( x \):
\[ (3x + 7y) - (3x + 2y) = 31 - 1 \]
This simplifies to:
\[ 7y - 2y = 30 \]
\[ 5y = 30 \]
Dividing both sides by 5 gives:
\[ y = 6 \]
Now that we have the value of \( y \), we can substitute it back into one of the original equations to find \( x \). We'll use Equation 1:
\[ 3x + 7(6) = 31 \]
This simplifies to:
\[ 3x + 42 = 31 \]
Subtracting 42 from both sides:
\[ 3x = 31 - 42 \]
\[ 3x = -11 \]
Dividing both sides by 3 gives:
\[ x = -\frac{11}{3} \quad \text{or} \quad -3 \frac{2}{3} \]
Therefore, the value of \( x \) is \( -3 \frac{2}{3} \).