Question

Eliminating a Variable

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system.

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

Eliminate the y-term from the third equation.

Short answer

The equivalent system for the given system of linear equations is$\left\{\begin{array}{l}x-3y+2z=-1\\ y+z=-1\\ z=-1\end{array}\right.$

Step-by-step solution

Step 1. Given information

We are given the set of linear equations that is-

$\left\{\begin{array}{r}x-3y+2z=-1\\ y+z=-1\\ 2y-z=1\end{array}\right.$

Step 2. Concept used

We have to eliminate y term from the third equation and then we have to write the new equivalent system.

Step 3. Calculation

Given the system of linear equations is

$\left\{\begin{array}{r}x-3y+2z=-1\\ y+z=-1\\ 2y-z=1\end{array}\right.$

For eliminating y term from the third equation which is can be done by replacing the third equation by adding $-2$ times the second equation to it as follows

$\begin{array}{l}-2\times [y+z=-1]\\ -2y-2z=\mathrm{2...}\left(i\right)\end{array}$

Third equation is

$2y-z=1...\left(ii\right)$

Adding (i) and (ii), we get

$(2y-2y)+(-2z-z)=(2+1)$

After solving, we get

$\begin{array}{l}-3z=3\\ z=-1\end{array}$

Hence, the system of linear equations become

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