Solve {c}{x+3y-z=a}{3x-y+z=b}{-2x+y-3z=c}

$$\left\{ \begin{array} { c } { x + 3 y - z = a } \\ { 3 x - y + z = b } \\ { - 2 x + y - 3 z = c } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{a+4b+c}{11}$$

$$y=\frac{7a-5b-4c}{22}$$

$$z=\frac{a-7b-10c}{22}$$

Short Steps Using Substitution

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

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

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

$$3\left(-3y+z+a\right)-y+z=b -2\left(-3y+z+a\right)+y-3z=c$$

Solve these equations for y and z respectively.

$$y=\frac{2}{5}z+\frac{3}{10}a-\frac{1}{10}b z=\frac{7}{5}y-\frac{2}{5}a-\frac{1}{5}c$$

Substitute \frac{2}{5}z+\frac{3}{10}a-\frac{1}{10}b for y in the equation z=\frac{7}{5}y-\frac{2}{5}a-\frac{1}{5}c.

$$z=\frac{7}{5}\left(\frac{2}{5}z+\frac{3}{10}a-\frac{1}{10}b\right)-\frac{2}{5}a-\frac{1}{5}c$$

Solve z=\frac{7}{5}\left(\frac{2}{5}z+\frac{3}{10}a-\frac{1}{10}b\right)-\frac{2}{5}a-\frac{1}{5}c for z.

$$z=\frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c$$

Substitute \frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c for z in the equation y=\frac{2}{5}z+\frac{3}{10}a-\frac{1}{10}b.

$$y=\frac{2}{5}\left(\frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c\right)+\frac{3}{10}a-\frac{1}{10}b$$

Calculate y from y=\frac{2}{5}\left(\frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c\right)+\frac{3}{10}a-\frac{1}{10}b.

$$y=\frac{7}{22}a-\frac{5}{22}b-\frac{2}{11}c$$

Substitute \frac{7}{22}a-\frac{5}{22}b-\frac{2}{11}c for y and \frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c for z in the equation x=-3y+z+a.

$$x=-3\left(\frac{7}{22}a-\frac{5}{22}b-\frac{2}{11}c\right)+\frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c+a$$

Calculate x from x=-3\left(\frac{7}{22}a-\frac{5}{22}b-\frac{2}{11}c\right)+\frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c+a.

$$x=\frac{1}{11}a+\frac{4}{11}b+\frac{1}{11}c$$

The system is now solved.

$$x=\frac{1}{11}a+\frac{4}{11}b+\frac{1}{11}c y=\frac{7}{22}a-\frac{5}{22}b-\frac{2}{11}c z=\frac{1}{22}a-\frac{7}{22}b-\frac{5}{11}c$$