Solve {l}{x+y+z=100}{8x+5y+z=502}{8x=5y}

$$\left\{ \begin{array} { l } { x + y + z = 100 } \\ { 8 x + 5 y + z = 502 } \\ { 8 x = 5 y } \end{array} \right.$$

Solve for x, y, z

$$x=30$$

$$y=48$$

$$z=22$$

Short Steps Using Substitution

Solve x+y+z=100 for x.

$$x=-y-z+100$$

Substitute -y-z+100 for x in the second and third equation.

$$8\left(-y-z+100\right)+5y+z=502 8\left(-y-z+100\right)=5y$$

Solve these equations for y and z respectively.

$$y=-\frac{7}{3}z+\frac{298}{3} z=100-\frac{13}{8}y$$

Substitute -\frac{7}{3}z+\frac{298}{3} for y in the equation z=100-\frac{13}{8}y.

$$z=100-\frac{13}{8}\left(-\frac{7}{3}z+\frac{298}{3}\right)$$

Solve z=100-\frac{13}{8}\left(-\frac{7}{3}z+\frac{298}{3}\right) for z.

$$z=22$$

Substitute 22 for z in the equation y=-\frac{7}{3}z+\frac{298}{3}.

$$y=-\frac{7}{3}\times 22+\frac{298}{3}$$

Calculate y from y=-\frac{7}{3}\times 22+\frac{298}{3}.

$$y=48$$

Substitute 48 for y and 22 for z in the equation x=-y-z+100.

$$x=-48-22+100$$

Calculate x from x=-48-22+100.

$$x=30$$

The system is now solved.

$$x=30 y=48 z=22$$