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

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. $$\begin{bmatrix} 1 & 0 & −1 & 3 & 9\\ 0 & 1& 2 & −5 & 8\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

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.