Question

Solve the system of equations using a matrix:$\left\{\begin{array}{l}x-y-z=1\\ -x+2y-3z=-4\\ 3x-2y-7z=0\end{array}\right.$

Short answer

The solution for the system of linear equation is $\left(5z-2,4z-3,z\right)$where, $z$is any real number.

Step-by-step solution

Step 1. Given information.

Consider the given system of equations,

$\left\{\begin{array}{l}x-y-z=1\\ -x+2y-3z=-4\\ 3x-2y-7z=0\end{array}\right.$

Step 2. Write in augmented form.

The augmented matrix for the given system of equations is

$\left[\begin{array}{cccr}1& -1& -1& 1\\ -1& 2& -3& -4\\ 3& -2& -7& 0\end{array}\right]$

Step 3. Apply row operations.

Apply $R_2+R_1\to R_2$and $R_3-3\times R_1\to R_3$,

$\left[\begin{array}{cccr}1& -1& -1& 1\\ 0& 1& -4& -3\\ 0& 1& -4& -3\end{array}\right]$

Apply $R_3-R_2\to R_3$,

$\left[\begin{array}{cccr}1& -1& -1& 1\\ 0& 1& -4& -3\\ 0& 0& 0& 0\end{array}\right]$

Step 4. Write in system of equations.

Writing the corresponding system of equations,

$$\begin{gathered} x-y-z=1......\left(i\right) \\ y-4z=-3......\left(ii\right) \end{gathered}$$

Solve equation (ii) for $y$,

$y=4z-3$

Substitute the value of $y$in equation (i),

$$\begin{gathered} x-\left(4z-3\right)-z=1 \\ x-5z+3=1 \\ x=5z-2 \end{gathered}$$

Step 5. Check the answers.

Substitute the values in equation (i),

$$\begin{gathered} \left(5z-2\right)-\left(4z-3\right)-z=1 \\ 5z-2-4z+3-z=1 \\ 1=1 \end{gathered}$$

This is true.