Question

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss–Jordan elimination. Show all your work.

$\left|\begin{array}{c}{x}_1-7x_2+x_5=3\\ x_3-2x_5=2\\ x_4+x_5=1\end{array}\right|$

Short answer

For the given system of equation $x_5,x_5.$are free variables and

$x_3=2+2x_{5,}{x}_4=1-x_{5,}=3+7x_2-x_5$.

Step-by-step solution

Write the augmented matrix

First of all we will make the augmented matrix to the corresponding system of equations.

$\left|\begin{array}{c}{x}_1-7x_2+x_5=3\\ x_3-2x_5=2\\ x_4+x_5=1\end{array}\right|$

The corresponding augmented matrix is written as:

$\left[\begin{array}{ccccc}1& -7& 0& 0& 1\\ 0& 0& 1& 0& -2\\ 0& 0& 0& 1& 1\end{array}\overline{)\begin{array}{c}3\\ 2\\ 1\end{array}}\right]$

Solving the Augmented matrix

The above augmented matrix obtained above is already in reduced form, then the

System of equation to the corresponding above matrix will be

$\left|\begin{array}{c}{x}_1=3+7x_2-x_5\\ x_3=2+2x_5\\ x_4=1-x_5\end{array}\right|$

Finding free variables

According to system $\left|\begin{array}{c}{x}_1=3+7x_2-x_5\\ x_3=2+2x_5\\ x_4=1-x_5\end{array}\right|$, the variables ${\mathrm{x}}_3,{\mathrm{x}}_4,{\mathrm{x}}_1$depend on ${\mathrm{x}}_2,{\mathrm{x}}_5.\mathrm{So},{\mathrm{x}}_2,{\mathrm{x}}_5.$are free variables.

Final answer

Thus, for the given system of equation${\mathrm{x}}_{2,}{\mathrm{x}}_5$ are free variables and ${\mathrm{x}}_3=2+2{\mathrm{x}}_5,{\mathrm{x}}_4=1-{\mathrm{x}}_5{\mathrm{x}}_1=3+7{\mathrm{x}}_2-{\mathrm{x}}_5$.