Question

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

$\left\{\begin{array}{l}2y+4z=-1\\ -2x+y+2z=-14\\ x-2y=0\end{array}\right\}$

Short answer

The solution set of the system $\left\{\begin{array}{l}2y+4z=-1\\ -2x+y+2z=-14\\ x-2y=0\end{array}\right\}$is $\left\{\left(\frac{1}{4},\frac{1}{2},\frac{-1}{2}\right)\right\}$.

Step-by-step solution

Step 1.Given information.

We are given a system of linear equations that is$\left\{\begin{array}{l}2y+4z=-1\\ -2x+y+2z=-14\\ x-2y=0\end{array}\right\}$

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$\left\{\begin{array}{l}2y+4z=-\mathrm{1......}\left(i\right)\\ -2x+y+2z=-\mathrm{14.........}\left(ii\right)\\ x-2y=\mathrm{0...............}\left(iii\right)\end{array}\right\}$

We can see that each equation is written in standard form,

$Ax+By+Cz=D$

We will change the order of the equations, to have x term in 1^st row ,

Thus, new system will be$\left\{\begin{array}{l}-2x+y+2z=-1\\ 2y+4z=-1\\ 4x-2y=0\end{array}\right.$

We will begin by eliminating x from equation (3),

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

$\begin{array}{l}equation\left(1\right)\\ \left\{\begin{array}{l}-2x+y+2z=-1\\ 2y+4z=-1\\ 4z=-2\end{array}\right.\end{array}$

Using equation

(3) we have,

$\begin{array}{l}4z=-2\\ z=-\frac{1}{2}\end{array}$

Now substituting value of z in equation (2) we have,

$\begin{array}{l}2y-2=-1\\ y=\frac{1}{2}\end{array}$

We will now substitute value of y,z in equation (1) we will have,

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

The solution set of the system is $\left\{\left(\frac{1}{4},\frac{1}{2},\frac{-1}{2}\right)\right\}$. Since the equation has a complete solution, the given system is consistent.