Question

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

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

Short answer

The solution set of the system$\left\{\begin{array}{l}x+z+2w=6\\ y-2z=-3\\ x+2y-z=-2\\ 2x+y+3z-2w=0\end{array}\right\}$hasinfinitely many solutions and the system is

Step-by-step solution

Step 1. Given information

The given system of equations is$\left\{\begin{array}{l}x+z+2w=6\\ y-2z=-3\\ x+2y-z=-2\\ 2x+y+3z-2w=0\end{array}\right\}$

Step 2. Find the complete solution of the given system of linear equations and show if it is inconsistent.

We will convert the system into triangular form that has the same solutions as the original system using gauss elimination method.

We will solve triangular form by using back substitution and check the consistency of the system.

$\begin{array}{l}x+z+2w=\mathrm{6........................}\left(1\right)\\ y-2z=-3\\ y=2z-\mathrm{3.................................}\left(2\right)\\ x+2y-z=-\mathrm{2..............................}\left(3\right)\\ 2x+y+3z-2w=\mathrm{0............................}\left(4\right)\end{array}$

from equations 2 and 3 we, get

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

Putting equation 5 in equation 1

$\begin{array}{l}1+2w=6\\ 2w=6-1\\ 2w=5\\ w=\frac{5}{2}\end{array}$

From equation 5 we, get

$x=1-z$

putting the value of x and w in equation 1 we, get

$\begin{array}{l}1-z+z+2\times \frac{5}{6}=6\\ 1+0+5=6\\ 6=6\end{array}$

since there is no variable left so there is no solution or (inconsistent)