Solve {l}{100x+150z=75}{100y+150z=100}{z-y-x=0}

$$\left\{ \begin{array} { l } { 100 x + 150 z = 75 } \\ { 100 y + 150 z = 100 } \\ { z - y - x = 0 } \end{array} \right.$$

Solve for x, z, y

$$x=\frac{3}{32}=0.09375$$

$$y=\frac{11}{32}=0.34375$$

$$z=\frac{7}{16}=0.4375$$

Short Steps Using Substitution

Reorder the equations.

$$z-y-x=0 100y+150z=100 100x+150z=75$$

Solve z-y-x=0 for x.

$$x=z-y$$

Substitute z-y for x in the equation 100x+150z=75.

$$100\left(z-y\right)+150z=75$$

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

$$z=\frac{2}{3}-\frac{2}{3}y y=\frac{5}{2}z-\frac{3}{4}$$

Substitute \frac{2}{3}-\frac{2}{3}y for z in the equation y=\frac{5}{2}z-\frac{3}{4}.

$$y=\frac{5}{2}\left(\frac{2}{3}-\frac{2}{3}y\right)-\frac{3}{4}$$

Solve y=\frac{5}{2}\left(\frac{2}{3}-\frac{2}{3}y\right)-\frac{3}{4} for y.

$$y=\frac{11}{32}$$

Substitute \frac{11}{32} for y in the equation z=\frac{2}{3}-\frac{2}{3}y.

$$z=\frac{2}{3}-\frac{2}{3}\times \frac{11}{32}$$

Calculate z from z=\frac{2}{3}-\frac{2}{3}\times \frac{11}{32}.

$$z=\frac{7}{16}$$

Substitute \frac{7}{16} for z and \frac{11}{32} for y in the equation x=z-y.

$$x=\frac{7}{16}-\frac{11}{32}$$

Calculate x from x=\frac{7}{16}-\frac{11}{32}.

$$x=\frac{3}{32}$$

The system is now solved.

$$x=\frac{3}{32} z=\frac{7}{16} y=\frac{11}{32}$$