Solve {l}{x+y+z=5}{x-2y+4z=13}{3y+4z=13}

$$\left\{ \begin{array}{l}{ x + y + z = 5 }\\{ x - 2 y + 4 z = 13 }\\{ 3 y + 4 z = 13 }\end{array} \right.$$

Solve for x, y, z

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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

$$z=3$$

Short Steps Using Substitution

Solve x+y+z=5 for x.

$$x=-y-z+5$$

Substitute -y-z+5 for x in the equation x-2y+4z=13.

$$-y-z+5-2y+4z=13$$

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

$$y=z-\frac{8}{3} z=-\frac{3}{4}y+\frac{13}{4}$$

Substitute z-\frac{8}{3} for y in the equation z=-\frac{3}{4}y+\frac{13}{4}.

$$z=-\frac{3}{4}\left(z-\frac{8}{3}\right)+\frac{13}{4}$$

Solve z=-\frac{3}{4}\left(z-\frac{8}{3}\right)+\frac{13}{4} for z.

$$z=3$$

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

$$y=3-\frac{8}{3}$$

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

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

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

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

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

$$x=\frac{5}{3}$$

The system is now solved.

$$x=\frac{5}{3} y=\frac{1}{3} z=3$$