Question-1

Consider the following system of linear equations:

\[ \begin{aligned} -2x_{1}+3x_{2}+x_{3} & =1\\ -x_{1}+x_{3} & =0\\ 2x_{2} & =5 \end{aligned} \]

• Convert the system into the form \(Ax=b\).

• Solve the system.

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We have:

\[ A=\begin{bmatrix}-2 & 3 & 1\\ -1 & 0 & 1\\ 0 & 2 & 0 \end{bmatrix},\,\,\,x=\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3} \end{bmatrix},\,\,\,b=\begin{bmatrix}1\\ 0\\ 5 \end{bmatrix} \]

From the last equation, we see that \(x_{2}=\frac{5}{2}\). The second equation shows that \(x_{1}=x_{3}\). Using these two facts in equation-(1), we get \(x_{1}=x_{3}=3x_{2}-1=\frac{13}{2}\). The solution is unique and is given by \(\left(\frac{13}{2},\frac{5}{2},\frac{13}{2}\right)\).