Question: Use the elimination method to solve the system of equations. \[ \begin{align*} 3x - 2y &= 27…

Use the elimination method to solve the system of equations.

\[ \begin{align*} 3x - 2y &= 27 \\ 5x + 7y &= -48 \end{align*} \]

The solution set is \(\boxed{\quad}\).

(Simplify your answer. Type an ordered pair.)

Solution

To solve the system of equations using the elimination method, we have: \[ \begin{cases} 3x - 2y = 27 & \\ 5x + 7y = -48 & \end{cases} \] First, we want to eliminate one of the variables. Let’s eliminate \(y\) by making the coefficients of \(y\) in both equations the same. Multiply the first equation by 7: \[ 7(3x - 2y) = 7(27) \] \[ 21x - 14y = 189 \] Multiply the second equation by 2: \[ 2(5x + 7y) = 2(-48) \] \[ 10x + 14y = -96 \] Now, add the two new equations: \[ 21x - 14y + 10x + 14y = 189 - 96 \] \[ 31x = 93 \] Solve for \(x\): \[ x = \frac{93}{31} \] \[ x = 3 \] Substitute \(x = 3\) into the first original equation to solve for \(y\): \[ 3(3) - 2y = 27 \] \[ 9 - 2y = 27 \] Subtract 9 from both sides: \[ -2y = 18 \] Divide by -2: \[ y = -9 \] The solution set is \((3, -9)\).