# 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}

Question
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. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$

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. $$\displaystyle{x}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{9}\backslash{8}\backslash{0}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{s}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash-{2}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{t}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{3}\backslash{5}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ And write the solution in vector form.

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