Jaya Legge

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. $\left[\begin{array}{ccccc}1& 0& -1& 3& 9\\ 0& 1& 2& -5& 8\\ 0& 0& 0& 0& 0\end{array}\right]$

Latisha Oneil

Expert

${x}_{1}-{s}_{3}+3{x}_{4}=9$
${x}_{2}+2{x}_{3}-5{x}_{4}=8$
$0=0$ 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. ${x}_{1}=9+{x}_{3}-3{x}_{4}$
${x}_{2}=8-2{x}_{3}+5{x}_{4}$
${x}_{3},\mathfrak{e}e$
${x}_{4},\mathfrak{e}e$ Solve for the leading entry for each individual equation. Determine the free variables, if any. ${x}_{1}=9+s-3t$
${x}_{2}=8-2s+5t$
${x}_{3}=s$
${x}_{4}=t$ Parameterize the free variables. $x=\left[\begin{array}{c}9\\ 8\\ 0\\ 0\end{array}\right]+s\left[\begin{array}{c}1\\ -2\\ 1\\ 0\end{array}\right]+t\left[\begin{array}{c}-3\\ 5\\ 0\\ 1\end{array}\right]$ And write the solution in vector form.

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