Question

Solve the system of equations using a matrix:$\left\{\begin{array}{l}2x+y=-4\\ x-y=-2\end{array}\right.$

Short answer

The solution for the system of linear equation is $\left(-2,0\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given system of equations,

$\left\{\begin{array}{l}2x+y=-4\\ x-y=-2\end{array}\right.$

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

$\left[\begin{array}{rrr}2& 1& -4\\ 1& -1& -2\end{array}\right]$

Step 3. Apply row operations.

Apply $\frac{R_1}{2}\to R_1$,

$\left[\begin{array}{rrr}1& \frac{1}{2}& -2\\ 1& -1& -2\end{array}\right]$

Apply $R_2-R_1\to R_2$,

$\left[\begin{array}{rrr}1& \frac{1}{2}& -2\\ 0& -\frac{3}{2}& 0\end{array}\right]$

Apply $\left(-\frac{2}{3}\right)\times R_2\to R_2$,

$\left[\begin{array}{rrr}1& \frac{1}{2}& -2\\ 0& 1& 0\end{array}\right]$

Now, the matrix is in row-echelon form.

Step 4. Write in system of equations.

Writing the corresponding system of equations,

$$\begin{gathered} x+\frac{1}{2}y=-2......\left(i\right) \\ y=0......\left(ii\right) \end{gathered}$$

Substitute $y$in equation (i),

$$\begin{gathered} x+\frac{1}{2}\times \left(0\right)=-2 \\ x=-2 \end{gathered}$$

Step 5. Check the answers.

Substitute the values $x,y$in equation (i),

$$\begin{gathered} -2+\frac{1}{2}\left(0\right)=-2 \\ -2+0=-2 \\ -2=-2 \end{gathered}$$

This is true.