Question

Solving a System of Equations in Three Variables

Find the complete solution of the linear system, or show that it is inconsistent.

$\left\{\begin{array}{l}x+y+z=0\\ -x+2y+5z=3\\ 3x-y=6\end{array}\right.$

Short answer

The solution for the given set of linear equations in three variables is $(x,y,z)=(1,-3,2)$

Step-by-step solution

Step 1. Given information

We are given the set of linear equations that is-

$\left\{\begin{array}{l}x+y+z=0\\ -x+2y+5z=3\\ 3x-y=6\end{array}\right.$

Step 2. Concept used

We need to find the complete solution of the given system of linear equations and show if it is inconsistent

We will eliminate variables using the gauss elimination method.

Then we will solve using back substitution and check the consistency of the system.

Step 3. Calculation

Given the system of linear equations is

$\left\{\begin{array}{l}x+y+z=0\\ -x+2y+5z=3\\ 3x-y=6\end{array}\right.$

We will replace the second equation by adding the first equation which gives

$\begin{array}{l}(-x+x)+(2y+y)+(5z+z)=3+0\\ y+2z=\mathrm{1.....}\left(i\right)\end{array}$

Now, we will eliminate x term from the third equation by adding $-3$times the first equation which results

$\begin{array}{l}(3x-3x)+(-y-3y)+(-3z)=6-0\\ -4y-3z=\mathrm{6...}\left(ii\right)\end{array}$

We will add $4$times equation (i) with equation (ii) which gives

$\begin{array}{l}(-4y+4y)+(-3z+8z)=6+4\\ 5z=10\\ z=2\end{array}$

We can write the system of linear equations as

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

We will substitute the value of z in the second equation which gives

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

Similarly, we will solve for the first equation which gives

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