Question: 9) \begin{cases} 4x - 5y = 9 \\ 5x - 6y = 9 \end{cases}

Solution

To solve the system of equations, we have: \[ \begin{cases} 4x - 5y = 9 & \\ 5x - 6y = 9 & \end{cases} \] First, let’s use the method of elimination. Multiply the first equation by 5 and the second equation by 4 to align the coefficients of \(x\). \[ \begin{cases} 5(4x - 5y) = 5(9) & \\ 4(5x - 6y) = 4(9) & \end{cases} \] This gives: \[ \begin{cases} 20x - 25y = 45 & \\ 20x - 24y = 36 & \end{cases} \] Subtract the second equation from the first: \[ (20x - 25y) - (20x - 24y) = 45 - 36 \] Simplify: \[ -25y + 24y = 9 \] \[ -y = 9 \] Divide by \(-1\): \[ y = -9 \] Now substitute \(y = -9\) back into the first equation: \[ 4x - 5(-9) = 9 \] Simplify: \[ 4x + 45 = 9 \] Subtract 45 from both sides: \[ 4x = 9 - 45 \] \[ 4x = -36 \] Divide by 4: \[ x = -9 \] The solution to the system of equations is \(x = -9\) and \(y = -9\).