Solve {l}{3x+2y+4z=5}{6x+3y-2z=2}{3x-4y+8z=5}

$$\left. \begin{array} { l } { 3 x + 2 y + 4 z = 5 } \\ { 6 x + 3 y - 2 z = 2 } \\ { 3 x - 4 y + 8 z = 5 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{1}{3}\approx 0.333333333$$

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

$$z=\frac{3}{4}=0.75$$

Short Steps Using Substitution

Solve 3x+2y+4z=5 for x.

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

Substitute -\frac{2}{3}y-\frac{4}{3}z+\frac{5}{3} for x in the second and third equation.

$$6\left(-\frac{2}{3}y-\frac{4}{3}z+\frac{5}{3}\right)+3y-2z=2 3\left(-\frac{2}{3}y-\frac{4}{3}z+\frac{5}{3}\right)-4y+8z=5$$

Solve these equations for y and z respectively.

$$y=8-10z z=\frac{3}{2}y$$

Substitute 8-10z for y in the equation z=\frac{3}{2}y.

$$z=\frac{3}{2}\left(8-10z\right)$$

Solve z=\frac{3}{2}\left(8-10z\right) for z.

$$z=\frac{3}{4}$$

Substitute \frac{3}{4} for z in the equation y=8-10z.

$$y=8-10\times \frac{3}{4}$$

Calculate y from y=8-10\times \frac{3}{4}.

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

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

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

Calculate x from x=-\frac{2}{3}\times \frac{1}{2}-\frac{4}{3}\times \frac{3}{4}+\frac{5}{3}.

$$x=\frac{1}{3}$$

The system is now solved.

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