Question: For each ordered pair, determine whether it is a solution to the system of equations. \[…

For each ordered pair, determine whether it is a solution to the system of equations.

\[ \begin{cases} -3x + 2y = 5 \\ 2x - 7y = 8 \end{cases} \]

\[ \begin{array}{c|c|c} (x, y) & \text{Is it a solution?} & \\ \hline (4, 0) & \text{Yes} & \text{No} \\ \hline (5, 4) & \text{Yes} & \text{No} \\ \hline (-7, -8) & \text{Yes} & \text{No} \\ \hline (-2, -3) & \text{Yes} & \text{No} \end{array} \]

Solution

For each ordered pair, we need to determine if it satisfies the system of equations: \[ \begin{cases} -3x + 2y = 5 & \\ 2x - 7y = 8 & \end{cases} \] Check (4, 0): Substitute \(x = 4\) and \(y = 0\) into the equations. First equation: \[ -3(4) + 2(0) = -12 + 0 = -12 \neq 5 \] Since \(-12 \neq 5\), \((4, 0)\) is not a solution. Check (5, 4): Substitute \(x = 5\) and \(y = 4\) into the equations. First equation: \[ -3(5) + 2(4) = -15 + 8 = -7 \neq 5 \] Since \(-7 \neq 5\), \((5, 4)\) is not a solution. Check (-7, -8): Substitute \(x = -7\) and \(y = -8\) into the equations. First equation: \[ -3(-7) + 2(-8) = 21 - 16 = 5 \] Second equation: \[ 2(-7) - 7(-8) = -14 + 56 = 42 \neq 8 \] Since \(42 \neq 8\), \((-7, -8)\) is not a solution. Check (-2, -3): Substitute \(x = -2\) and \(y = -3\) into the equations. First equation: \[ -3(-2) + 2(-3) = 6 - 6 = 0 \neq 5 \] Since \(0 \neq 5\), \((-2, -3)\) is not a solution. In conclusion, none of the given ordered pairs satisfy both equations in the system.