Solve {l}{x+13y-2z=0}{2x-z=3}{y=5}

$$\left\{ \begin{array}{l}{ x + 13 y - 2 z = 0 }\\{ 2 x - z = 3 }\\{ y = 5 }\end{array} \right.$$

Solve for x, y, z

$$x = \frac{71}{3} = 23\frac{2}{3} \approx 23.666666667$$

$$y=5$$

$$z = \frac{133}{3} = 44\frac{1}{3} \approx 44.333333333$$

Short Steps Using Substitution

Reorder the equations.

$$y=5 2x-z=3 x+13y-2z=0$$

Substitute 5 for y in the equation x+13y-2z=0.

$$x+13\times 5-2z=0$$

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

$$x=\frac{3}{2}+\frac{1}{2}z z=\frac{65}{2}+\frac{1}{2}x$$

Substitute \frac{3}{2}+\frac{1}{2}z for x in the equation z=\frac{65}{2}+\frac{1}{2}x.

$$z=\frac{65}{2}+\frac{1}{2}\left(\frac{3}{2}+\frac{1}{2}z\right)$$

Solve z=\frac{65}{2}+\frac{1}{2}\left(\frac{3}{2}+\frac{1}{2}z\right) for z.

$$z=\frac{133}{3}$$

Substitute \frac{133}{3} for z in the equation x=\frac{3}{2}+\frac{1}{2}z.

$$x=\frac{3}{2}+\frac{1}{2}\times \frac{133}{3}$$

Calculate x from x=\frac{3}{2}+\frac{1}{2}\times \frac{133}{3}.

$$x=\frac{71}{3}$$

The system is now solved.

$$x=\frac{71}{3} y=5 z=\frac{133}{3}$$