Questions: A linear function is given. Complete parts (a)-(d).

Solution Steps

We start with the system of linear equations: \[ \begin{align_} 2x + 3y &= 5 \\ 4x - y &= 1 \end{align_} \] We can represent this system in augmented matrix form \( [A | b] \): \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 2 & 3 & 5 \\ 4 & -1 & 1 \\ \end{array} \right] \]

Next, we perform row operations to simplify the augmented matrix. We can divide the first row by 2: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{3}{2} & \frac{5}{2} \\ 4 & -1 & 1 \\ \end{array} \right] \] Then, we eliminate the first variable from the second row by subtracting 4 times the first row from the second row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{3}{2} & \frac{5}{2} \\ 0 & -7 & -9 \\ \end{array} \right] \]

Next, we can simplify the second row by dividing it by -7: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{3}{2} & \frac{5}{2} \\ 0 & 1 & \frac{9}{7} \\ \end{array} \right] \] Now, we eliminate the second variable from the first row by subtracting \(\frac{3}{2}\) times the second row from the first row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & 0 & \frac{4}{7} \\ 0 & 1 & \frac{9}{7} \\ \end{array} \right] \]

From the final augmented matrix, we can read the solutions directly: \[ x = \frac{4}{7}, \quad y = \frac{9}{7} \]

Final Answer