Question

solve the following using Gaussian Elimination method 2x + y + 3z = 4 x + y + 2z = 0 2x + 4y + 6z = -8

Answer 1

$\begin{cases} 2x + y + 3z = 4 \\ x + y + 2z = 0 \\ 2x + 4y + 6z = -8 \end{cases} \rightarrow \begin{pmatrix} 2& 1& 3& 4\\ 1& 1& 2& 0\\ 2& 4& 6& -8 \end{pmatrix} \sim \begin{pmatrix} 2& 1& 3& 4\\ 0& \frac{1}{2}& \frac{1}{2}& -2\\ 0& 3& 3& -12 \end{pmatrix} \sim \begin{pmatrix} 2& 1& 3& 4\\ 0& \frac{1}{2}& \frac{1}{2}& -2\\ 0& 0& 0& 0 \end{pmatrix} \sim \begin{pmatrix} 1& \frac{1}{2}& \frac{3}{2}& 2\\ 0& 1& 1& -4\\ 0& 0& 0& 0 \end{pmatrix} \Rightarrow \begin{cases} x + \frac{y}{2} + \frac{3z}{2} = 2 \\ y + z = -4 \end{cases} \Rightarrow \begin{cases} x = 2 - \frac{y}{2} - \frac{3z}{2} \\ y = -4 - z \end{cases} \Rightarrow \begin{cases} x = 4 - z \\ y = -4 - z \end{cases}z = a\Rightarrow Answer:\begin{cases} x = 4 - a \\ y = -4 - a \\ z = a \end{cases}$