# Solve the system of equations using matrices. Use the Gaussian elimination method with​ back-substitution. x+4y=0 x+5y+z=1 5x-y-z=79

Solve the system of equations using matrices. Use the Gaussian elimination method with​ back-substitution.
x+4y=0 x+5y+z=1 5x-y-z=79
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Mitchel Aguirre
Step 1
x+4y=0
x+5y+z=1
5x-y-z=79
The above system can be written matrix form
$\left[\begin{array}{cccc}1& 4& 0& 0\\ 1& 5& 1& 1\\ 5& -1& -1& 79\end{array}\right]$
${R}_{2}\to {R}_{2}-{R}_{1}$
${R}_{3}\to {R}_{3}-5{R}_{1}$
$\left[\begin{array}{cccc}1& 4& 0& 0\\ 0& 1& 1& 1\\ 0& -21& -1& 79\end{array}\right]$
${R}_{3}\to {R}_{3}+21{R}_{2}$
$\left[\begin{array}{cccc}1& 4& 0& 0\\ 0& 1& 1& 1\\ 0& 0& 20& 100\end{array}\right]$
$⇒x+4y=0$
y+z=1
20z=100
Using back substitution
$z=\frac{100}{20}$
z=5
y=1-5
y=-4
x+4y=0
$⇒x=-4\left(-4\right)$
x=16
Jeffrey Jordon