Questions: Solve the following systems of equations for the unknown variables. Enter your answer in (x, y) format. 6 x+3 y=18 5 x+y=12

Solution Steps

We start with the system of equations: \[ \begin{array}{l} 6x + 3y = 18 \\ 5x + y = 12 \end{array} \] We can represent this system in augmented matrix form \( [A | b] \): \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 6 & 3 & 18 \\ 5 & 1 & 12 \\ \end{array} \right] \]

Next, we perform row operations to simplify the augmented matrix. We can start by dividing the first row by 6: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{1}{2} & 3 \\ 5 & 1 & 12 \\ \end{array} \right] \] Then, we eliminate the first variable from the second row by subtracting 5 times the first row from the second row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{1}{2} & 3 \\ 0 & -\frac{3}{2} & -3 \\ \end{array} \right] \]

Next, we can simplify the second row by multiplying it by \(-\frac{2}{3}\): \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & \frac{1}{2} & 3 \\ 0 & 1 & 2 \\ \end{array} \right] \]

Now, we can eliminate the second variable from the first row by subtracting \(\frac{1}{2}\) times the second row from the first row: \[ \left[ A | b \right] = \left[ \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 2 \\ \end{array} \right] \]

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

Final Answer