Solve {l}{x+2y+4z=17}{-3x+2y-z=-2}{5y+y-2z=1}

$$\left\{ \begin{array} { l } { x + 2 y + 4 z = 17 } \\ { - 3 x + 2 y - z = - 2 } \\ { 5 y + y - 2 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{16}{41}\approx 0.390243902$$

$$y = \frac{109}{82} = 1\frac{27}{82} \approx 1.329268293$$

$$z = \frac{143}{41} = 3\frac{20}{41} \approx 3.487804878$$

Short Steps Using Substitution

Solve x+2y+4z=17 for x.

$$x=-2y-4z+17$$

Substitute -2y-4z+17 for x in the equation -3x+2y-z=-2.

$$-3\left(-2y-4z+17\right)+2y-z=-2$$

Solve the second equation for y and the third equation for z.

$$y=-\frac{11}{8}z+\frac{49}{8} z=3y-\frac{1}{2}$$

Substitute -\frac{11}{8}z+\frac{49}{8} for y in the equation z=3y-\frac{1}{2}.

$$z=3\left(-\frac{11}{8}z+\frac{49}{8}\right)-\frac{1}{2}$$

Solve z=3\left(-\frac{11}{8}z+\frac{49}{8}\right)-\frac{1}{2} for z.

$$z=\frac{143}{41}$$

Substitute \frac{143}{41} for z in the equation y=-\frac{11}{8}z+\frac{49}{8}.

$$y=-\frac{11}{8}\times \frac{143}{41}+\frac{49}{8}$$

Calculate y from y=-\frac{11}{8}\times \frac{143}{41}+\frac{49}{8}.

$$y=\frac{109}{82}$$

Substitute \frac{109}{82} for y and \frac{143}{41} for z in the equation x=-2y-4z+17.

$$x=-2\times \frac{109}{82}-4\times \frac{143}{41}+17$$

Calculate x from x=-2\times \frac{109}{82}-4\times \frac{143}{41}+17.

$$x=\frac{16}{41}$$

The system is now solved.

$$x=\frac{16}{41} y=\frac{109}{82} z=\frac{143}{41}$$