Questions: REASONING Find the values of a and b so that the solution of the linear system is (-9,1). ax+by=-31 Equation 1 ax-by=-41 Equation 2

Solution Steps

Given the solution point \((-9, 1)\), we can substitute \(x = -9\) and \(y = 1\) into the equations:

\[ \begin{align_} a(-9) + b(1) &= -31 \quad \text{(Equation 1)} \\ a(-9) - b(1) &= -41 \quad \text{(Equation 2)} \end{align_} \]

This leads to the following system of equations:

\[ \begin{align_} -9a + b &= -31 \\ -9a - b &= -41 \end{align_} \]

We can represent the system of equations in augmented matrix form:

\[ \left[ A | b \right] = \left[ \begin{array}{cc|c} -9 & 1 & -31 \\ -9 & -1 & -41 \\ \end{array} \right] \]

We perform row operations to simplify the augmented matrix. The first step is to make the leading coefficient of the first row equal to 1:

\[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & -\frac{1}{9} & \frac{31}{9} \\ -9 & -1 & -41 \\ \end{array} \right] \]

Next, we eliminate the first column of the second row:

\[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & -\frac{1}{9} & \frac{31}{9} \\ 0 & -2 & -10 \\ \end{array} \right] \]

Now, we can make the leading coefficient of the second row equal to 1:

\[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & -\frac{1}{9} & \frac{31}{9} \\ 0 & 1 & 5 \\ \end{array} \right] \]

We can now perform back substitution to find the values of \(a\) and \(b\). From the second row, we have:

\[ b = 5 \]

Substituting \(b\) back into the first row:

\[ a - \frac{1}{9}(5) = \frac{31}{9} \]

This simplifies to:

\[ a = 4 \]

The final values obtained from the system of equations are:

\[ a = 4, \quad b = 5 \]

Final Answer