Solve {l}{2x-y-z=1}{2x-3y-4z=0}{x+y-z=4}

$$\left. \begin{array} { l } { 2 x - y - z = 1 } \\ { 2 x - 3 y - 4 z = 0 } \\ { x + y - z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=-1$$

Short Steps Using Substitution

Solve 2x-y-z=1 for y.

$$y=2x-z-1$$

Substitute 2x-z-1 for y in the second and third equation.

$$2x-3\left(2x-z-1\right)-4z=0 x+2x-z-1-z=4$$

Solve these equations for x and z respectively.

$$x=\frac{3}{4}-\frac{1}{4}z z=\frac{3}{2}x-\frac{5}{2}$$

Substitute \frac{3}{4}-\frac{1}{4}z for x in the equation z=\frac{3}{2}x-\frac{5}{2}.

$$z=\frac{3}{2}\left(\frac{3}{4}-\frac{1}{4}z\right)-\frac{5}{2}$$

Solve z=\frac{3}{2}\left(\frac{3}{4}-\frac{1}{4}z\right)-\frac{5}{2} for z.

$$z=-1$$

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

$$x=\frac{3}{4}-\frac{1}{4}\left(-1\right)$$

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

$$x=1$$

Substitute 1 for x and -1 for z in the equation y=2x-z-1.

$$y=2\times 1-\left(-1\right)-1$$

Calculate y from y=2\times 1-\left(-1\right)-1.

$$y=2$$

The system is now solved.

$$x=1 y=2 z=-1$$