Questions: Listen USING TOOLS Use a graphing calculator to solve the system of linear equations. 4 x-y=1.5 2 x+y=1.5

Solution Steps

We start with the system of linear equations: \[ \begin{array}{l} 4x - y = 1.5 \\ 2x + y = 1.5 \end{array} \] We can represent this system in augmented matrix form \( [A | b] \): \[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 4 & -1 & 1.5 \\ 2 & 1 & 1.5 \\ \end{array} \right] \]

We perform row operations to simplify the augmented matrix. First, we can divide the first row by 4: \[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & -\frac{1}{4} & \frac{3}{8} \\ 2 & 1 & 1.5 \\ \end{array} \right] \] Next, we eliminate the first element of the second row by subtracting 2 times the first row from the second row: \[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & -\frac{1}{4} & \frac{3}{8} \\ 0 & \frac{3}{2} & \frac{3}{4} \\ \end{array} \right] \]

Now, we simplify the second row by dividing it by \(\frac{3}{2}\): \[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & -\frac{1}{4} & \frac{3}{8} \\ 0 & 1 & \frac{1}{2} \\ \end{array} \right] \] Next, we eliminate the second element of the first row by adding \(\frac{1}{4}\) times the second row to the first row: \[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & 0 & \frac{1}{2} \\ 0 & 1 & \frac{1}{2} \\ \end{array} \right] \]

From the final augmented matrix, we can read the solutions directly: \[ x = \frac{1}{2}, \quad y = \frac{1}{2} \]

The solution to the system of equations is: \[ \begin{cases} x = \frac{1}{2} \\ y = \frac{1}{2} \end{cases} \]

Final Answer