Question: (\[ \begin{align*} 10) & \quad 5x - 7y = 19 \\ & \quad -4x + 5y = -17 \end{align*} \]\)…

\(\[ \begin{align*} 10) & \quad 5x - 7y = 19 \\ & \quad -4x + 5y = -17 \end{align*} \]\)

Solution

To solve the system of equations, we can use the elimination method. The system of equations given is: \[ \begin{cases} 5x - 7y = 19 & \\ -4x + 5y = -17 & \end{cases} \] Multiply the first equation by 5 and the second equation by 7 to eliminate \( y \). \[ \begin{cases} 25x - 35y = 95 & \\ -28x + 35y = -119 & \end{cases} \] Add the two equations together: \[ 25x - 35y + (-28x + 35y) = 95 + (-119) \] Simplify: \[ -3x = -24 \] Divide by \(-3\): \[ x = 8 \] Now substitute \( x = 8 \) into the first equation: \[ 5(8) - 7y = 19 \] Simplify: \[ 40 - 7y = 19 \] Subtract 40 from both sides: \[ -7y = -21 \] Divide by \(-7\): \[ y = 3 \] The solution to the system of equations is: \( x = 8 \) and \( y = 3 \)