Questions: Follow the directions to solve the system of equations by elimination. 8x + 7y = 39 4x - 14y = -68 1. Multiply the first equation to enable the elimination of the y-term. 2. Add the equations to eliminate the y-terms. 3. Solve the new equation for the x-value. 4. Substitute the x-value back into either original equation to find the y-value. 5. Check the solution. The solution to the system of equations is ( . ).

Solution Steps

To enable the elimination of the \( y \)-term, we multiply the first equation \( 8x + 7y = 39 \) by \( 2 \): \[ 16x + 14y = 78 \]

The updated system of equations is: \[ \begin{array}{l} 16x + 14y = 78 \\ 4x - 14y = -68 \end{array} \]

We can represent the system in matrix form and apply Gaussian elimination: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 16 & 14 & 78 \\ 4 & -14 & -68 \\ \end{array} \right] \] After performing row operations, we arrive at: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{7}{8} & \frac{39}{8} \\ 0 & 1 & 5 \\ \end{array} \right] \] From this, we can deduce: \[ x = \frac{1}{2}, \quad y = 5 \]

We substitute \( x = \frac{1}{2} \) and \( y = 5 \) back into the original equations to verify:

1. For the first equation: \[ 8 \left(\frac{1}{2}\right) + 7(5) = 4 + 35 = 39 \]
2. For the second equation: \[ 4 \left(\frac{1}{2}\right) - 14(5) = 2 - 70 = -68 \] Both equations are satisfied, confirming the solution is correct.

Final Answer