Question

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

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

Short answer

The solution set of the system$\left\{\begin{array}{r}2x+3y-z=1\\ x+2y=3\\ x+3y+z=4\end{array}\right.$is $\left\{\right(-3,3,2\left)\right\}$ and the system is consistent.

Step-by-step solution

Step 1. Given information.

The given system of equations is $\left\{\begin{array}{r}2x+3y-z=1\\ x+2y=3\\ x+3y+z=4\end{array}\right.$and show if it is inconsistent

Step 2. 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.

The given system of equations is$\left\{\begin{array}{l}2x+3y-z=\mathrm{1............}\left(1\right)\\ x+2y=\mathrm{3................}\left(2\right)\\ x+3y+z=\mathrm{4.............}\left(3\right)\end{array}\right\}$

We can see that each equation is written in standard form,$Ax+By+Cz=D$

We will begin by eliminating x from equation 2 ,

$\left\{\begin{array}{ll}2x+3y-z=1& \dots \dots \left(1\right)\\ -y-z=-5& \dots \dots \left(2\right)\\ x+3y+z=4& \dots \dots .\left(3\right)\end{array}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}equation}1-2\times \text{equation}2\right.$

Now we will eliminate x from equation 3 ,

$\left\{\begin{array}{ll}2x+3y-z=1& \dots \dots \left(1\right)\\ -y-z=-5& \dots \dots \left(2\right)\\ -3y+z=-7& \dots \dots \left(3\right)\end{array}\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}equation}1-2\times \text{equation}3\right.$

Now we will eliminate y from equation 3 ,

$\left\{\begin{array}{lll}2x+3y-z=1& \dots \dots .\left(1\right)& 3\times \text{equation}2-\text{equation}3\\ -y-z=-5& \dots \dots .\left(2\right)& \\ -4z=-8& \dots \mathrm{..}\left(3\right)& \end{array}\right.$

Using equation 3 we have,

$\begin{array}{l}-4z=-8\\ z=2\end{array}$

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

$\begin{array}{c}-y-z=-5\\ -y-2=-5\\ y=3\end{array}$

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

$2x+3y-z=1$

$\begin{array}{l}2x+9-2=1\\ x=-3\end{array}$

The solution set of the system is $\left\{\right(-3,3,2\left)\right\}$since we have a complete solution, the given system is consistent.