Question

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

$\left\{\begin{array}{r}x-2y-3z=5\\ 2x+y-z=5\\ 4x-3y-7z=5\end{array}\right.$

Short answer

The given system$\left\{\begin{array}{r}x-2y-3z=5\\ 2x+y-z=5\\ 4x-3y-7z=5\end{array}\right.$has no solution and it is inconsistent.

Step-by-step solution

Step 1.Given information.

We are given a system of linear equations that is$\left\{\begin{array}{r}x-2y-3z=5\\ 2x+y-z=5\\ 4x-3y-7z=5\end{array}\right.$

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$\left\{\begin{array}{l}x-2y-3z=\mathrm{5.............}\left(1\right)\\ 2x+y-z=\mathrm{5...................}\left(2\right)\\ 4x-3y-7z=\mathrm{5..............}\left(3\right)\end{array}\right\}$

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

$Ax+By+Cz=D$

We will begin by eliminating x from equation (2),

$\left\{\begin{array}{ll}x-2y-3z=5& \dots \dots \left(1\right)\\ -5y-5z=5& \dots \mathrm{..}\left(2\right)\\ 4x-3y-7z=5& \dots \mathrm{..}\left(3\right)\end{array}2\times \right.\text{equation1-equation2}$

Now we will eliminate x from equation 3 ,

$\left\{\begin{array}{ll}x-2y-3z=5& \dots \dots \left(1\right)\\ -5y-5z=5& \dots \dots \left(2\right)\\ -5y-5z=15& \dots \dots \left(3\right)\end{array}4\times \text{equation}1-\text{equation}3\right.$

Now we will eliminate y from equation 3 ,

$\left\{\begin{array}{ll}x-2y-3z=5& \dots \dots \left(1\right)\\ -5y-5z=5& \dots \dots \left(2\right)\\ 0=-10& \dots \dots \left(3\right)\end{array}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}equation}2-\text{equation}3\right.$

The system is now in triangular form, but the third equation says 0=-10 which is false. No matter what values we assign to x, y, z the third equation will never be true. This means that the system has no solution and it is inconsistent.