Question

In the following exercises, solve each system of equations using a matrix.

$$\begin{gathered} x+y=3z=-1 \\ y-z=0 \\ -x+2y=1 \end{gathered}$$

Short answer

The solution for the system of linear equations is (4z - 1, - z, z)

here z is any real number.

Step-by-step solution

Step 1. Given information

The three given equations are:

$$\begin{gathered} x+y-3z=-1 \\ y-z=0 \\ -x+2y=1 \end{gathered}$$

Step 2. Argumented form

First, create an augmented version of this system.

The first equation provides us the first row, the second equation gives us the second row, and the third equation gives us the third row in the augmented matrix. The equal signs are changed by a vertical line.

The argumented matrix is

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

Step 3. Calculation

Consider the argumented matrix:

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

By applying row operations

By applying row operations:

$$\begin{gathered} \left[\begin{array}{rrrr}1& 1& -3& -1\\ 0& 1& -1& 0\\ -1& 2& 0& 1\end{array}\right]\stackrel{R_3\to R_1+R_3}{\to }\left[\begin{array}{rrrr}1& 1& -3& -1\\ 0& 1& -1& 0\\ 0& 3& -3& 0\end{array}\right] \\ \stackrel{R_3\to \frac{1}{3}{R}_3}{\to }\left[\begin{array}{rrrr}1& 1& -3& -1\\ 0& 1& -1& 0\\ 0& 1& -1& 0\end{array}\right] \\ \stackrel{R_3\to R_2-R_3}{\to }\left[\begin{array}{rrrr}1& 1& -3& -1\\ 0& 1& -1& 0\\ 0& 0& 0& 0\end{array}\right] \end{gathered}$$

The row echelon form is $\left|\begin{array}{cccc}1& 1& -3& -1\\ 0& 1& -1& 0\\ 0& 0& -0& 0\end{array}\right|$

The corresponding equations are:

$$\begin{gathered} x+y-3z=-1 \\ -y-z=0 \\ 0=0 \end{gathered}$$

Because the sentence 0=0 is true.

This, like when we solved through substitution, indicates that we are dealing with a dependent system. There are an unlimited number of options.

In the second equation, solve for y in terms of z.

$$\begin{gathered} -y-z=0 \\ -y=z \\ y=-z \end{gathered}$$

$$\begin{gathered} \mathrm{Substitute}y=-z \\ \mathrm{in}\mathrm{first}\mathrm{equation}x+y-3z=-1 \\ x+y-3z=-1 \\ x-z-3z=-1 \\ x=4z-1 \\ x=4z-1 \end{gathered}$$

Hence ,The solution for the system of linear equations is (4z - 1, - z, z)

here z is any real number.