Solve {ccc}{0}&{-3}&{-3}{-3}&{-2}&{7}{-3}&{1}&{1}{l}{x}{y}{z}={c}{3}{-1}{a}

$$\left( \begin{array} { c c c } { 0 } & { - 3 } & { - 3 } \\ { - 3 } & { - 2 } & { 7 } \\ { - 3 } & { 1 } & { 1 } \end{array} \right) \left( \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right) = \left( \begin{array} { c } { 3 } \\ { - 1 } \\ { a } \end{array} \right)$$

Solve for x, y, z

$$x=\frac{-a-1}{3}$$

$$y=\frac{a-5}{9}$$

$$z=\frac{-a-4}{9}$$

Short Steps Using Substitution

Reorder the equations.

$$-3x+y+z=a -3x-2y+7z=-1 -3y-3z=3$$

Solve -3x+y+z=a for y.

$$y=3x-z+a$$

Substitute 3x-z+a for y in the second and third equation.

$$-3x-2\left(3x-z+a\right)+7z=-1 -3\left(3x-z+a\right)-3z=3$$

Solve these equations for z and x respectively.

$$z=x+\frac{2}{9}a-\frac{1}{9} x=-\frac{1}{3}-\frac{1}{3}a$$

Substitute -\frac{1}{3}-\frac{1}{3}a for x in the equation z=x+\frac{2}{9}a-\frac{1}{9}.

$$z=-\frac{1}{3}-\frac{1}{3}a+\frac{2}{9}a-\frac{1}{9}$$

Calculate z from z=-\frac{1}{3}-\frac{1}{3}a+\frac{2}{9}a-\frac{1}{9}.

$$z=-\frac{4}{9}-\frac{1}{9}a$$

Substitute -\frac{4}{9}-\frac{1}{9}a for z and -\frac{1}{3}-\frac{1}{3}a for x in the equation y=3x-z+a.

$$y=3\left(-\frac{1}{3}-\frac{1}{3}a\right)-\left(-\frac{4}{9}-\frac{1}{9}a\right)+a$$

Calculate y from y=3\left(-\frac{1}{3}-\frac{1}{3}a\right)-\left(-\frac{4}{9}-\frac{1}{9}a\right)+a.

$$y=-\frac{5}{9}+\frac{1}{9}a$$

The system is now solved.

$$x=-\frac{1}{3}-\frac{1}{3}a y=-\frac{5}{9}+\frac{1}{9}a z=-\frac{4}{9}-\frac{1}{9}a$$