Questions: Solve the system of equations using substitution. 3x - 2y = 3 -4x + y = 6 (x, y) = d Select Choice

Solution Steps

We start with the given system of equations: \[ \begin{array}{l} 3x - 2y = 3 \quad (1) \\ -4x + y = 6 \quad (2) \end{array} \]

We can represent the system in augmented matrix form \( [A | b] \): \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 3 & -2 & 3 \\ -4 & 1 & 6 \\ \end{array} \right] \]

We perform row operations to reduce the matrix to row echelon form. The steps are as follows:

1. Divide the first row by 3: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 1 \\ -4 & 1 & 6 \\ \end{array} \right] \]

2. Add 4 times the first row to the second row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 1 \\ 0 & -\frac{5}{3} & 10 \\ \end{array} \right] \]

3. Multiply the second row by \(-\frac{3}{5}\): \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 1 \\ 0 & 1 & -6 \\ \end{array} \right] \]

4. Add \(\frac{2}{3}\) times the second row to the first row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & 0 & -3 \\ 0 & 1 & -6 \\ \end{array} \right] \]

From the final row echelon form, we can read the solutions: \[ x = -3 \\ y = -6 \]

Final Answer