Question: Solve the system by elimination. If the system is inconsistent or has dependent equations, say so…

Solve the system by elimination. If the system is inconsistent or has dependent equations, say so.

\[ \begin{align*} 8x + 3y &= 10 \\ -8x + y &= 14 \end{align*} \]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The system has a single solution. The solution set is \(\boxed{\quad}\). (Type an ordered pair. Type integers or simplified fractions.)

B. There are infinitely many solutions and the equations are dependent. The solution set is \(\{(x, y) | 8x + 3y = 10\}\).

C. The system is inconsistent. The solution set is the empty set.

Solution

To solve the system of equations by elimination: \[ 8x + 3y = 10 \] \[ -8x + y = 14 \] Step 1: Eliminate \( x \) by adding the two equations. \[ \begin{aligned} 8x + 3y &= 10 \\ -8x + y &= 14 \\ \hline 0x + 4y &= 24 \\ \end{aligned} \] Step 2: Solve for \( y \). \[ 4y = 24 \] \[ y = 6 \] Step 3: Substitute \( y = 6 \) into the first equation to solve for \( x \). \[ 8x + 3(6) = 10 \] \[ 8x + 18 = 10 \] \[ 8x = -8 \] \[ x = -1 \] The solution to the system is \( (-1, 6) \). The correct choice is: A. The system has a single solution. The solution set is { \((-1, 6)\) }.