Solve {l}{x+y+z=2}{x+y+T+w=3z}{T+2w+x=y}

$$\left. \begin{array} { l } { x + y + z = 2 } \\ { x + y + T + w = 3 z } \\ { T + 2 w + x = y } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{5T}{8}-\frac{9w}{8}+\frac{3}{4}$$

$$y=\frac{3T}{8}+\frac{7w}{8}+\frac{3}{4}$$

$$z=\frac{T}{4}+\frac{w}{4}+\frac{1}{2}$$

Short Steps Using Substitution

Reorder the equations.

$$T+2w+x=y x+y+T+w=3z x+y+z=2$$

Solve T+2w+x=y for y.

$$y=T+2w+x$$

Substitute T+2w+x for y in the second and third equation.

$$x+T+2w+x+T+w=3z x+T+2w+x+z=2$$

Solve these equations for x and z respectively.

$$x=-T-\frac{3}{2}w+\frac{3}{2}z z=-2x-T-2w+2$$

Substitute -T-\frac{3}{2}w+\frac{3}{2}z for x in the equation z=-2x-T-2w+2.

$$z=-2\left(-T-\frac{3}{2}w+\frac{3}{2}z\right)-T-2w+2$$

Solve z=-2\left(-T-\frac{3}{2}w+\frac{3}{2}z\right)-T-2w+2 for z.

$$z=\frac{1}{2}+\frac{1}{4}T+\frac{1}{4}w$$

Substitute \frac{1}{2}+\frac{1}{4}T+\frac{1}{4}w for z in the equation x=-T-\frac{3}{2}w+\frac{3}{2}z.

$$x=-T-\frac{3}{2}w+\frac{3}{2}\left(\frac{1}{2}+\frac{1}{4}T+\frac{1}{4}w\right)$$

Calculate x from x=-T-\frac{3}{2}w+\frac{3}{2}\left(\frac{1}{2}+\frac{1}{4}T+\frac{1}{4}w\right).

$$x=\frac{3}{4}-\frac{5}{8}T-\frac{9}{8}w$$

Substitute \frac{3}{4}-\frac{5}{8}T-\frac{9}{8}w for x in the equation y=T+2w+x.

$$y=T+2w+\frac{3}{4}-\frac{5}{8}T-\frac{9}{8}w$$

Calculate y from y=T+2w+\frac{3}{4}-\frac{5}{8}T-\frac{9}{8}w.

$$y=\frac{3}{4}+\frac{3}{8}T+\frac{7}{8}w$$

The system is now solved.

$$x=\frac{3}{4}-\frac{5}{8}T-\frac{9}{8}w y=\frac{3}{4}+\frac{3}{8}T+\frac{7}{8}w z=\frac{1}{2}+\frac{1}{4}T+\frac{1}{4}w$$