# Use the​ Gauss-Jordan method to solve the system of equations. If the system has infinitely many​ solutions, give the solution with z arbitrary.x-5y+2z=13x-4y+2z=-1

Use the​ Gauss-Jordan method to solve the system of equations. If the system has infinitely many​ solutions, give the solution with z arbitrary.
$x-5y+2z=1$
$3x-4y+2z=-1$

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Step 1
To find the solution of system of equations reduce the matrix $\left[A\mid B\right]$ where A is the matrix formed by the coefficients of LHS of the equations and B is the matrix formed by the RHS of the equations.
Step 2
Find the matrix $\left[A\mid B\right]$ from the given system of equations and row reduce it.
$\left[\begin{array}{cccc}1& -5& 2& 1\\ 3& -4& 2& -1\end{array}\right]{R}_{2}\to {R}_{2}-3{R}_{1}$
$\left[\begin{array}{cccc}1& -5& 2& 1\\ 0& 11& -4& -4\end{array}\right]{R}_{2}\to \frac{{R}_{2}}{11}$
$\left[\begin{array}{cccc}1& -5& 2& 1\\ 0& 1& -\frac{4}{11}& -\frac{4}{11}\end{array}\right]{R}_{1}\to {R}_{1}+5{R}_{2}$
$\left[\begin{array}{cccc}1& 0& \frac{2}{11}& -\frac{9}{11}\\ 0& 1& -\frac{4}{11}& -\frac{4}{11}\end{array}\right]$
Step 3
Form the equation from the above matrix.
$x+\frac{2}{11}z=-\frac{9}{11}$
$y-\frac{4}{11}z=-\frac{4}{11}$
Step 4
Let z=t. Find the value of x and y.
$x+\frac{2}{11}\left(t\right)=-\frac{9}{11}$
$x=-\frac{9}{11}-\frac{2t}{11}$
$y-\frac{4}{11}\left(t\right)=-\frac{4}{11}$
$y=-\frac{4}{11}+\frac{4t}{11}$
Step 5
Answer: For random $z=t$ the solution set will be,
$\left\{\left(\frac{-9}{11}-\frac{2t}{11},\frac{-4}{11}+\frac{4t}{11}\right),t\right\}$