Questions: Consider the following system of two linear equations: 3x - 2y = 12 3x + 2y = 0 Select the graph that correctly displays this system of equations and point of intersection.

Solution Steps

We start with the given system of linear equations: \[ \begin{array}{l} 3x - 2y = 12 \\ 3x + 2y = 0 \end{array} \]

The system can be represented in augmented matrix form \( [A | b] \): \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 3 & -2 & 12 \\ 3 & 2 & 0 \\ \end{array} \right] \]

We perform Gaussian elimination to solve the system. The steps are as follows:

1. Divide the first row by 3: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 4 \\ 3 & 2 & 0 \\ \end{array} \right] \]

2. Subtract 3 times the first row from the second row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 4 \\ 0 & 4 & -12 \\ \end{array} \right] \]

3. Divide the second row by 4: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & -\frac{2}{3} & 4 \\ 0 & 1 & -3 \\ \end{array} \right] \]

4. Substitute back to eliminate \( y \) from the first row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & -3 \\ \end{array} \right] \]

From the final augmented matrix, we can read the solutions: \[ x = 2, \quad y = -3 \]

Final Answer