Solve {l}{x+y-z=-2}{2x+y+z=5}{3x+y+z=4}

$$\left\{ \begin{array} { l } { x + y - z = - 2 } \\ { 2 x + y + z = 5 } \\ { 3 x + y + z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=3$$

$$z=4$$

Short Steps Using Substitution

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

$$x=-y+z-2$$

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

$$2\left(-y+z-2\right)+y+z=5 3\left(-y+z-2\right)+y+z=4$$

Solve these equations for y and z respectively.

$$y=-9+3z z=\frac{5}{2}+\frac{1}{2}y$$

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

$$z=\frac{5}{2}+\frac{1}{2}\left(-9+3z\right)$$

Solve z=\frac{5}{2}+\frac{1}{2}\left(-9+3z\right) for z.

$$z=4$$

Substitute 4 for z in the equation y=-9+3z.

$$y=-9+3\times 4$$

Calculate y from y=-9+3\times 4.

$$y=3$$

Substitute 3 for y and 4 for z in the equation x=-y+z-2.

$$x=-3+4-2$$

Calculate x from x=-3+4-2.

$$x=-1$$

The system is now solved.

$$x=-1 y=3 z=4$$