Solve {l}{x+y+z=21}{10x=2*15y}{10x=2*12z}

$$\left. \begin{array} { l } { x + y + z = 21 } \\ { 10 x = 2 \times 15 y } \\ { 10 x = 2 \times 12 z } \end{array} \right.$$

Solve for x, y, z

$$x=12$$

$$y=4$$

$$z=5$$

Short Steps Using Substitution

Solve x+y+z=21 for x.

$$x=-y-z+21$$

Substitute -y-z+21 for x in the second and third equation.

$$10\left(-y-z+21\right)=2\times 15y 10\left(-y-z+21\right)=2\times 12z$$

Solve these equations for y and z respectively.

$$y=\frac{21}{4}-\frac{1}{4}z z=\frac{105}{17}-\frac{5}{17}y$$

Substitute \frac{21}{4}-\frac{1}{4}z for y in the equation z=\frac{105}{17}-\frac{5}{17}y.

$$z=\frac{105}{17}-\frac{5}{17}\left(\frac{21}{4}-\frac{1}{4}z\right)$$

Solve z=\frac{105}{17}-\frac{5}{17}\left(\frac{21}{4}-\frac{1}{4}z\right) for z.

$$z=5$$

Substitute 5 for z in the equation y=\frac{21}{4}-\frac{1}{4}z.

$$y=\frac{21}{4}-\frac{1}{4}\times 5$$

Calculate y from y=\frac{21}{4}-\frac{1}{4}\times 5.

$$y=4$$

Substitute 4 for y and 5 for z in the equation x=-y-z+21.

$$x=-4-5+21$$

Calculate x from x=-4-5+21.

$$x=12$$

The system is now solved.

$$x=12 y=4 z=5$$