Question: Solve the following system by elimination. \[ \begin{align*} x - \frac{1}{5}y &= 2 \\ -x +…

Solve the following system by elimination.

\[ \begin{align*} x - \frac{1}{5}y &= 2 \\ -x + \frac{5}{6}y &= -\frac{25}{3} \end{align*} \]

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

A. The solution set is \(\boxed{\quad}\). (Type an ordered pair.)

B. The system has dependent equations. The solution set is \(\{(x, y) | 5x - y = 10\}\).

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

Solution

The problem requires solving the system of equations by elimination: \[ \begin{cases} x - \frac{1}{5}y = 2 & \\ -\frac{5}{6}x + y = -\frac{25}{3} & \end{cases} \] First, clear the fractions by multiplying the first equation by 5 and the second equation by 6 to eliminate the fractions. The first equation becomes: \[ 5x - y = 10 \] The second equation becomes: \[ -5x + 6y = -50 \] Now we add the equations to eliminate \(x\): \[ \begin{align*} 5x - y & = 10 \\ -5x + 6y & = -50 \\ \hline 0x + 5y & = -40 \end{align*} \] Divide the entire equation by 5: \[ y = -8 \] Substitute \(y = -8\) back into the first simplified equation: \[ 5x - (-8) = 10 \] \[ 5x + 8 = 10 \] Subtract 8 from both sides: \[ 5x = 2 \] Divide by 5: \[ x = \frac{2}{5} \] The solution set is \( \left(\frac{2}{5}, -8\right) \). Therefore, the correct choice is A. The solution set is \(\left\{\left(\frac{2}{5}, -8\right)\right\}\).