Question-7

Solve the following system using Gaussian elimination.

\[ \begin{aligned} 2x_{1}+x_{2} & =3\\ x_{1}+3x_{2} & =4 \end{aligned} \]

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Let us form the augmented matrix:

\[ \begin{bmatrix}2 & 1 & & 3\\ 1 & 3 & & 4 \end{bmatrix} \]

We now start row reduction:

\(R_{1}\leftrightarrow R_{2}\)

\[ \begin{bmatrix}1 & 3 & & 4\\ 2 & 1 & & 3 \end{bmatrix} \]

\(R_{2}\rightarrow R_{2}-2R_{1}\)

\[ \begin{bmatrix}1 & 3 & & 4\\ 0 & -5 & & -5 \end{bmatrix} \]

\(R_{2}\rightarrow\cfrac{-1}{5}R_{2}\)

\[ \begin{bmatrix}1 & 3 & & 4\\ 0 & 1 & & 1 \end{bmatrix} \]

The matrix is now in REF. We now proceed to get the RREF.

\(R_{1}\rightarrow R_{1}-3R_{2}\)

\[ \begin{bmatrix}\boldsymbol{1} & 0 & & 1\\ 0 & \boldsymbol{1} & & 1 \end{bmatrix} \]

We have transformed \(Ax=b\) into \(Rx=c\), where \(R\) is in RREF. All that remains is to read off the solution here:

\[ \begin{aligned} x_{2} & =1\\ x_{1} & =1 \end{aligned} \]

Since both \(x_{1}\) and \(x_{2}\) are dependent variables, the solution is unique.