Question

Solving a System of Equations in Three Variables Find the complete solution of the linear system, or show that it is inconsistent.

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

Short answer

The solution for the given set of linear equations in three variables is

$(x,y,z)=(0,1,2)$

Step-by-step solution

Step 1.Given information.

We are given a system of linear equations that is$\left\{\begin{array}{l}2x+4y-z=2\\ x+2y-3z=-4\\ 3x-y+z=1\end{array}\right\}$

Step 2. 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.

Given a system of linear equations is$\left\{\begin{array}{l}2x+4y-z=2\\ x+2y-3z=-4\\ 3x-y+z=1\end{array}\right\}$

We will replace the second equation by adding -0.5 times the first equation which gives

$\begin{array}{c}(x-x)+(2y-2y)+(-3z+0.5z)=-4-1\\ -2.5z=-5\\ z=2\end{array}$

We will try to eliminate x term from the third equation by adding -3 times the first equation which results

$\begin{array}{c}(3x-3x)+(-y-6y)+(z+9z)=1+12\\ -7y+10z=13\end{array}$

Now, we can write the linear equations as

$\left\{\begin{array}{r}2x+4y-z=2\\ z=2\\ -7y+10z=13\end{array}\right\}$

Step 3. substitute the value in the third equation which gives

$\begin{array}{c}-7y+10z=13\\ -7y+10\left(2\right)=13\\ -7y=-7\\ y=1\end{array}$

Similarly, for the first equation, we can write as

$\begin{array}{c}2x+4y-z=2\\ 2x+4-2=2\\ 2x=0\\ x=0\end{array}$