Question

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system

Forms of linear equations
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asked 2021-01-16

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. \(\begin{bmatrix} 1 & 0 & −1 & 3 & 9\\ 0 & 1& 2 & −5 & 8\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\)

Expert Answers (1)

2021-01-17

\(\displaystyle{x}_{{{1}}}-{s}_{{{3}}}+{3}{x}_{{{4}}}={9}\)
\(\displaystyle{x}_{{{2}}}+{2}{x}_{{{3}}}-{5}{x}_{{{4}}}={8}\)
\(\displaystyle{0}={0}\) The augmented matrix does not contain a row in which the only nonzero entry appears in the last column. Therefore, this system of equations must be consistent. Convert the augmented matrix into a system of equations. \(\displaystyle{x}_{{1}}={9}+{x}_{{3}}-{3}{x}_{{4}}\)
\(\displaystyle{x}_{{2}}={8}-{2}{x}_{{3}}+{5}{x}_{{4}}\)
\(\displaystyle{x}_{{3}},{\mathfrak{{e}}}{e}\)
\(\displaystyle{x}_{{4}},{\mathfrak{{e}}}{e}\) Solve for the leading entry for each individual equation. Determine the free variables, if any. \(\displaystyle{x}_{{1}}={9}+{s}-{3}{t}\)
\(\displaystyle{x}_{{2}}={8}-{2}{s}+{5}{t}\)
\(\displaystyle{x}_{{3}}={s}\)
\(\displaystyle{x}_{{4}}={t}\) Parameterize the free variables. \(x = \begin{bmatrix}9 \\ 8\\0\\0 \end{bmatrix} + s \begin{bmatrix}1 \\-2\\1\\0 \end{bmatrix} + t \begin{bmatrix}-3 \\ 5\\0\\1 \end{bmatrix}\) And write the solution in vector form.

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