Solve {l}{x+y-3z+6=2c}{3x-y+z-t=2a}{-x+3y-z+t=2b}

$$\left\{ \begin{array}{l}{ x + y - 3 z + 6 = 2 c }\\{ 3 x - y + z - t = 2 a }\\{ - x + 3 y - z + t = 2 b }\end{array} \right.$$

Solve for x, y, z

$$x=\frac{b}{6}+\frac{c}{6}+\frac{t}{4}+\frac{2a}{3}-\frac{1}{2}$$

$$y=\frac{a}{3}+\frac{5b}{6}-\frac{c}{6}-\frac{t}{4}+\frac{1}{2}$$

$$z=\frac{a}{3}+\frac{b}{3}-\frac{2c}{3}+2$$

Short Steps Using Substitution

Solve x+y-3z+6=2c for x.

$$x=-y+3z-6+2c$$

Substitute -y+3z-6+2c for x in the second and third equation.

$$3\left(-y+3z-6+2c\right)-y+z-t=2a -\left(-y+3z-6+2c\right)+3y-z+t=2b$$

Solve these equations for y and z respectively.

$$y=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a z=y-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t$$

Substitute \frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a for y in the equation z=y-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t.

$$z=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t$$

Solve z=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a-\frac{1}{2}b+\frac{3}{2}-\frac{1}{2}c+\frac{1}{4}t for z.

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

Substitute 2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b for z in the equation y=\frac{5}{2}z-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a.

$$y=\frac{5}{2}\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a$$

Calculate y from y=\frac{5}{2}\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-\frac{9}{2}+\frac{3}{2}c-\frac{1}{4}t-\frac{1}{2}a.

$$y=\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b$$

Substitute \frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b for y and 2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b for z in the equation x=-y+3z-6+2c.

$$x=-\left(\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b\right)+3\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-6+2c$$

Calculate x from x=-\left(\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b\right)+3\left(2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b\right)-6+2c.

$$x=-\frac{1}{2}+\frac{1}{6}c+\frac{1}{4}t+\frac{2}{3}a+\frac{1}{6}b$$

The system is now solved.

$$x=-\frac{1}{2}+\frac{1}{6}c+\frac{1}{4}t+\frac{2}{3}a+\frac{1}{6}b y=\frac{1}{2}-\frac{1}{6}c-\frac{1}{4}t+\frac{1}{3}a+\frac{5}{6}b z=2-\frac{2}{3}c+\frac{1}{3}a+\frac{1}{3}b$$