Solve {c}{2x+3y-5z=-4}{4x-y+3z=2}{3x+2y+4z=1}

$$\left\{ \begin{array} { c } { 2 x + 3 y - 5 z = - 4 } \\ { 4 x - y + 3 z = 2 } \\ { 3 x + 2 y + 4 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

$$y=-\frac{1}{2}=-0.5$$

$$z=\frac{1}{2}=0.5$$

Short Steps Using Substitution

Reorder the equations.

$$4x-y+3z=2 2x+3y-5z=-4 3x+2y+4z=1$$

Solve 4x-y+3z=2 for y.

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

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

$$2x+3\left(4x+3z-2\right)-5z=-4 3x+2\left(4x+3z-2\right)+4z=1$$

Solve these equations for x and z respectively.

$$x=\frac{1}{7}-\frac{2}{7}z z=-\frac{11}{10}x+\frac{1}{2}$$

Substitute \frac{1}{7}-\frac{2}{7}z for x in the equation z=-\frac{11}{10}x+\frac{1}{2}.

$$z=-\frac{11}{10}\left(\frac{1}{7}-\frac{2}{7}z\right)+\frac{1}{2}$$

Solve z=-\frac{11}{10}\left(\frac{1}{7}-\frac{2}{7}z\right)+\frac{1}{2} for z.

$$z=\frac{1}{2}$$

Substitute \frac{1}{2} for z in the equation x=\frac{1}{7}-\frac{2}{7}z.

$$x=\frac{1}{7}-\frac{2}{7}\times \frac{1}{2}$$

Calculate x from x=\frac{1}{7}-\frac{2}{7}\times \frac{1}{2}.

$$x=0$$

Substitute 0 for x and \frac{1}{2} for z in the equation y=4x+3z-2.

$$y=4\times 0+3\times \frac{1}{2}-2$$

Calculate y from y=4\times 0+3\times \frac{1}{2}-2.

$$y=-\frac{1}{2}$$

The system is now solved.

$$x=0 y=-\frac{1}{2} z=\frac{1}{2}$$