Problem-1

system of equations

Question

\[ \begin{equation*} \begin{aligned} x_{1} +2x_{2} -x_{3} & =0\\ x_{2} -x_{3} & =2\\ x_{2} +x_{3} & =4 \end{aligned} \end{equation*} \]

This system is represented as \(Ax=b\). Find \(A\), \(x\) and \(b\). Solve for \(x\). How many solutions does this system have?

Hint

Use equations \((2)\) and \((3)\) to eliminate \(x_2\) or \(x_3\).

Solution

\[ \begin{equation*} A=\begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1\\ 0 & 1 & 1 \end{bmatrix} ,\ x=\begin{bmatrix} x_{1}\\ x_{2}\\ x_{3} \end{bmatrix} ,\ \ b=\begin{bmatrix} 0\\ 2\\ 4 \end{bmatrix} \end{equation*} \]

Adding equations \((2)\) and \((3)\) and solving for \(x_2\), we get \(x_2 = 3\). Substituting this back in equation \((3)\) gives \(x_3 = 1\). Finally, plugging \(x_2 = 3, x_3 = 1\) into equation \((1)\) gives \(x_1 = -5\). \(\begin{bmatrix} -5\\ 3\\ 1 \end{bmatrix}\) is the only solution to this system.