Question

# Use Gauss-Jordan row reduction to solve the given system of equations. 9x−10y =9 36x−40y=36

Equations
Use Gauss-Jordan row reduction to solve the given system of equations.
9x−10y =9
36x−40y=36

2021-02-10
Step 1
The objective is to find a solution for the given system of equations using Gauss-Jordan elimination method.
The system of equations given is
9x−10y =9
36x−40y=36
The matrix representation of the above system of equations is
$$\displaystyle{\left[\begin{array}{cc} {9}&{10}\\{36}&{40}\end{array}\right]}{\left[\begin{array}{c} {x}\\{y}\end{array}\right]}={\left[\begin{array}{c} {9}\\{36}\end{array}\right]},{A}{X}={B}$$.
Then the augmented matrix $$\displaystyle{\left[{A}{\mid}{B}\right]}$$ becomes
$$\displaystyle{\left[\begin{array}{ccc} {9}&{10}&{9}\\{36}&{40}&{36}\end{array}\right]}$$
Step 2
Now, reducing the matrix $$\displaystyle{\left[{A}{\mid}{B}\right]}$$ step-by-step,(Gauss Jordan row reduction),
$$\displaystyle{R}_{{2}}\rightarrow\frac{{1}}{{4}}{R}_{{2}}\Rightarrow$$ The matrix $$\displaystyle{\left[{A}{\mid}{B}\right]}$$ becomes $$\displaystyle{\left[\begin{array}{ccc} {9}&{10}&{9}\\{9}&{10}&{9}\end{array}\right]}$$
$$\displaystyle{R}_{{2}}\rightarrow{R}_{{2}}-{R}_{{1}}\Rightarrow$$ The matrix $$\displaystyle{\left[{A}{\mid}{B}\right]}$$ becomes $$\displaystyle{\left[\begin{array}{ccc} {9}&{10}&{9}\\{0}&{0}&{0}\end{array}\right]}$$
The reduced matrix can be expressed as 9x+10y = 9.
Let y = k, any arbitrary value, the $$\displaystyle{x}=\frac{{{9}-{10}{k}}}{{9}}$$
This system has infinitely many solutions.
$$\displaystyle{\left({x},{y}\right)}={\left(\frac{{{9}-{10}{k}}}{{9}},{k}\right)}$$