Write the solution of the given homogeneous system in parametric

luipieduq3 2021-11-14 Answered
Write the solution of the given homogeneous system in parametric vector form
\(\displaystyle{3}{x}_{{1}}+{3}{x}_{{2}}+{6}{x}_{{3}}={0}\)
\(\displaystyle-{9}{x}_{{1}}-{9}{x}_{{2}}-{18}{x}_{{3}}={0}\)
\(\displaystyle-{6}{x}_{{2}}-{6}{x}_{{3}}={0}\)
where the solution set is \[x=\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\]

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Expert Answer

Elizabeth Witte
Answered 2021-11-15 Author has 7169 answers
To describe the system of equation in the parametric vector form, solve for \(\displaystyle{x}_{{1}},\ {x}_{{2}}\) and \(\displaystyle{x}_{{3}}\).
\(\displaystyle{3}{x}_{{1}}+{3}{x}_{{2}}+{6}{x}_{{3}}={0}\)
\(\displaystyle-{9}{x}_{{1}}-{9}{x}_{{2}}-{18}{x}_{{3}}={0}\)
\(\displaystyle-{6}{x}_{{2}}-{6}{x}_{{3}}={0}\)
Let Solving equation we get:
\(\displaystyle-{6}{x}_{{2}}={6}{x}_{{3}}\)
\(\displaystyle{x}_{{2}}=-{x}_{{3}}\)
Substitute \(\displaystyle{x}_{{2}}=-{x}_{{3}}\) in equation, we get:
\(\displaystyle{3}{x}_{{1}}-{3}{x}_{{3}}+{6}{x}_{{3}}={0}\)
\(\displaystyle{3}{x}_{{1}}=-{3}{x}_{{3}}\)
\(\displaystyle{x}_{{1}}=-{x}_{{3}}\)
Thus, the paranetric vector form for system of equation is as follows:
\[x=\begin{bmatrix}-1\\-1\\1\end{bmatrix}\]
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