Question

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

Answer 1

Step 1
x+4y=0
x+5y+z=1
5x-y-z=79
The above system can be written matrix form
\(\begin{bmatrix}1&4&0&0 \\1&5&1&1\\5&-1&-1&79 \end{bmatrix}\)
\(R_2 \rightarrow R_2-R_1\)
\(R_3 \rightarrow R_3-5R_1\)
\(\begin{bmatrix}1&4&0&0 \\0&1&1&1\\0&-21&-1&79 \end{bmatrix}\)
\(R_3 \rightarrow R_3+21R_2\)
\(\begin{bmatrix}1&4&0&0 \\0&1&1&1\\0&0&20&100 \end{bmatrix}\)
\(\Rightarrow x+4y=0\)
y+z=1
20z=100
Using back substitution
\(z=\frac{100}{20}\)
z=5
y=1-5
y=-4
x+4y=0
\(\Rightarrow x=-4(-4)\)
x=16