Solve {l}{15x-3y+2z=1}{3x+4y-z=-18}{z=13}

$$\left. \begin{array} { l } { 15 x - 3 y + 2 z = 1 } \\ { 3 x + 4 y - z = - 18 } \\ { z = 13 } \end{array} \right.$$

Solve for x, y, z

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

$$y=0$$

$$z=13$$

Short Steps Using Substitution

Reorder the equations.

$$z=13 3x+4y-z=-18 15x-3y+2z=1$$

Substitute 13 for z in the second and third equation.

$$3x+4y-13=-18 15x-3y+2\times 13=1$$

Solve these equations for y and x respectively.

$$y=-\frac{3}{4}x-\frac{5}{4} x=\frac{1}{5}y-\frac{5}{3}$$

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

$$x=\frac{1}{5}\left(-\frac{3}{4}x-\frac{5}{4}\right)-\frac{5}{3}$$

Solve x=\frac{1}{5}\left(-\frac{3}{4}x-\frac{5}{4}\right)-\frac{5}{3} for x.

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

Substitute -\frac{5}{3} for x in the equation y=-\frac{3}{4}x-\frac{5}{4}.

$$y=-\frac{3}{4}\left(-\frac{5}{3}\right)-\frac{5}{4}$$

Calculate y from y=-\frac{3}{4}\left(-\frac{5}{3}\right)-\frac{5}{4}.

$$y=0$$

The system is now solved.

$$x=-\frac{5}{3} y=0 z=13$$