 # Use Gauss-Jordan row reduction to solve the given system of equations. 9x−10y =9 36x−40y=36 Falak Kinney 2021-02-09 Answered
Use Gauss-Jordan row reduction to solve the given system of equations.
9x−10y =9
36x−40y=36
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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
$\left[\begin{array}{cc}9& 10\\ 36& 40\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}9\\ 36\end{array}\right],AX=B$.
Then the augmented matrix $\left[A\mid B\right]$ becomes
$\left[\begin{array}{ccc}9& 10& 9\\ 36& 40& 36\end{array}\right]$
Step 2
Now, reducing the matrix $\left[A\mid B\right]$ step-by-step,(Gauss Jordan row reduction),
${R}_{2}\to \frac{1}{4}{R}_{2}⇒$ The matrix $\left[A\mid B\right]$ becomes $\left[\begin{array}{ccc}9& 10& 9\\ 9& 10& 9\end{array}\right]$
${R}_{2}\to {R}_{2}-{R}_{1}⇒$ The matrix $\left[A\mid B\right]$ becomes $\left[\begin{array}{ccc}9& 10& 9\\ 0& 0& 0\end{array}\right]$
The reduced matrix can be expressed as 9x+10y = 9.
Let y = k, any arbitrary value, the $x=\frac{9-10k}{9}$
This system has infinitely many solutions.
$\left(x,y\right)=\left(\frac{9-10k}{9},k\right)$