Solve {l}{x+y+z=352}{z=x/4}{x-y=z-2}

$$\left.\begin{array} { l } { x + y + z = 352 } \\ { z = \frac { x } { 4 } } \\ { x - y = z - 2 } \end{array} \right\}$$

Solve for x, y, z

$$x=175$$

$$y = \frac{533}{4} = 133\frac{1}{4} = 133.25$$

$$z = \frac{175}{4} = 43\frac{3}{4} = 43.75$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$x+y+z=352 4z=x x-y=z-2$$

Reorder the equations.

$$4z=x x+y+z=352 x-y=z-2$$

Solve 4z=x for x.

$$x=4z$$

Substitute 4z for x in the second and third equation.

$$4z+y+z=352 4z-y=z-2$$

Solve these equations for y and z respectively.

$$y=-5z+352 z=-\frac{2}{3}+\frac{1}{3}y$$

Substitute -5z+352 for y in the equation z=-\frac{2}{3}+\frac{1}{3}y.

$$z=-\frac{2}{3}+\frac{1}{3}\left(-5z+352\right)$$

Solve z=-\frac{2}{3}+\frac{1}{3}\left(-5z+352\right) for z.

$$z=\frac{175}{4}$$

Substitute \frac{175}{4} for z in the equation y=-5z+352.

$$y=-5\times \frac{175}{4}+352$$

Calculate y from y=-5\times \frac{175}{4}+352.

$$y=\frac{533}{4}$$

Substitute \frac{175}{4} for z in the equation x=4z.

$$x=4\times \frac{175}{4}$$

Calculate x from x=4\times \frac{175}{4}.

$$x=175$$

The system is now solved.

$$x=175 y=\frac{533}{4} z=\frac{175}{4}$$