Solve {l}{4x+3y+2z=1}{3x-2y+3z=9}{x-y+z=21}

$$\left. \begin{array} { l } { 4 x + 3 y + 2 z = 1 } \\ { 3 x - 2 y + 3 z = 9 } \\ { x - y + z = 21 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{229}{2} = 114\frac{1}{2} = 114.5$$

$$y=-54$$

$$z = -\frac{295}{2} = -147\frac{1}{2} = -147.5$$

Short Steps Using Substitution

Reorder the equations.

$$x-y+z=21 3x-2y+3z=9 4x+3y+2z=1$$

Solve x-y+z=21 for x.

$$x=y-z+21$$

Substitute y-z+21 for x in the second and third equation.

$$3\left(y-z+21\right)-2y+3z=9 4\left(y-z+21\right)+3y+2z=1$$

Solve these equations for y and z respectively.

$$y=-54 z=\frac{7}{2}y+\frac{83}{2}$$

Substitute -54 for y in the equation z=\frac{7}{2}y+\frac{83}{2}.

$$z=\frac{7}{2}\left(-54\right)+\frac{83}{2}$$

Calculate z from z=\frac{7}{2}\left(-54\right)+\frac{83}{2}.

$$z=-\frac{295}{2}$$

Substitute -54 for y and -\frac{295}{2} for z in the equation x=y-z+21.

$$x=-54-\left(-\frac{295}{2}\right)+21$$

Calculate x from x=-54-\left(-\frac{295}{2}\right)+21.

$$x=\frac{229}{2}$$

The system is now solved.

$$x=\frac{229}{2} y=-54 z=-\frac{295}{2}$$