Questions: Solve the system by the substitution method. x+y = -2 y = -2x Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution.

Solution Steps

We start with the given system of equations: \[ \begin{aligned} x + y & = -2 \quad \text{(1)} \\ y & = -2x \quad \text{(2)} \end{aligned} \] We can rearrange equation (2) to standard form: \[ 2x + y = 0 \quad \text{(3)} \]

The augmented matrix for the system of equations (1) and (3) is: \[ \left[ A | b \right] = \left[ \begin{array}{cc|c} 1 & 1 & -2 \\ 2 & 1 & 0 \\ \end{array} \right] \]

We perform row operations to reduce the augmented matrix:

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

From the final augmented matrix, we can read the solutions: \[ \begin{aligned} x & = 2 \\ y & = -4 \end{aligned} \]

Final Answer