Solve {c}{x-y-2z=10}{y-3z=-4}{3x-y+5z=21}

$$\left\{ \begin{array} { c } { x - y - 2 z = 10 } \\ { y - 3 z = - 4 } \\ { 3 x - y + 5 z = 21 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{97}{17} = 5\frac{12}{17} \approx 5.705882353$$

$$y = -\frac{71}{17} = -4\frac{3}{17} \approx -4.176470588$$

$$z=-\frac{1}{17}\approx -0.058823529$$

Short Steps Using Substitution

Solve x-y-2z=10 for x.

$$x=y+2z+10$$

Substitute y+2z+10 for x in the equation 3x-y+5z=21.

$$3\left(y+2z+10\right)-y+5z=21$$

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

$$y=3z-4 z=-\frac{2}{11}y-\frac{9}{11}$$

Substitute 3z-4 for y in the equation z=-\frac{2}{11}y-\frac{9}{11}.

$$z=-\frac{2}{11}\left(3z-4\right)-\frac{9}{11}$$

Solve z=-\frac{2}{11}\left(3z-4\right)-\frac{9}{11} for z.

$$z=-\frac{1}{17}$$

Substitute -\frac{1}{17} for z in the equation y=3z-4.

$$y=3\left(-\frac{1}{17}\right)-4$$

Calculate y from y=3\left(-\frac{1}{17}\right)-4.

$$y=-\frac{71}{17}$$

Substitute -\frac{71}{17} for y and -\frac{1}{17} for z in the equation x=y+2z+10.

$$x=-\frac{71}{17}+2\left(-\frac{1}{17}\right)+10$$

Calculate x from x=-\frac{71}{17}+2\left(-\frac{1}{17}\right)+10.

$$x=\frac{97}{17}$$

The system is now solved.

$$x=\frac{97}{17} y=-\frac{71}{17} z=-\frac{1}{17}$$