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

\(\[ \begin{cases} 4x - 5y = 9 \\ 5x - 6y = 9 \end{cases} \]\)

Solution

The image shows a system of equations: \[ \begin{cases} 5(4x - 5y = 9) & \\ 4(5x - 6y = 9) & \end{cases} \] First, distribute the numbers in front of the parentheses in each equation. For the first equation: \(5(4x - 5y = 9)\) Multiply each term by 5: \(5 \cdot 4x - 5 \cdot 5y = 5 \cdot 9\) Which becomes: \[ 20x - 25y = 45 \] For the second equation: \(4(5x - 6y = 9)\) Multiply each term by 4: \(4 \cdot 5x - 4 \cdot 6y = 4 \cdot 9\) Which becomes: \[ 20x - 24y = 36 \] Now, the system of equations is: \[ \begin{cases} 20x - 25y = 45 & \\ 20x - 24y = 36 & \end{cases} \] Subtract the second equation from the first to eliminate \(x\): \((20x - 25y) - (20x - 24y) = 45 - 36\) This simplifies to: \(-25y + 24y = 9\) \(-y = 9\) Solve for \(y\): \[ y = -9 \] Substitute \(y = -9\) back into the first equation to solve for \(x\): \(20x - 25(-9) = 45\) Simplify: \(20x + 225 = 45\) Subtract 225 from both sides: \(20x = 45 - 225\) \(20x = -180\) Divide by 20: \[ x = -9 \] The solution is: \[ (x, y) = (-9, -9) \]