Solve {c}{0,6x+0,5y=0,3(x+y+z)}{0,2x+0,6y+0,6z=x+y+z/2}{y=100+x}

$$\left\{ \begin{array} { c } { 0,6 x + 0,5 y = 0,3 ( x + y + z ) } \\ { 0,2 x + 0,6 y + 0,6 z = \frac { x + y + z } { 2 } } \\ { y = 100 + x } \end{array} \right.$$

Solve for x, y, z

$$x=500$$

$$y=600$$

$$z=900$$

Short Steps Using Substitution

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

$$0,6x+0,5y=0,3\left(x+y+z\right) 0,4x+1,2y+1,2z=x+y+z y=100+x$$

Reorder the equations.

$$y=100+x 0,4x+1,2y+1,2z=x+y+z 0,6x+0,5y=0,3\left(x+y+z\right)$$

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

$$0,4x+1,2\left(100+x\right)+1,2z=x+100+x+z 0,6x+0,5\left(100+x\right)=0,3\left(x+100+x+z\right)$$

Solve these equations for x and z respectively.

$$x=50+0,5z z=\frac{200}{3}+\frac{5}{3}x$$

Substitute 50+0,5z for x in the equation z=\frac{200}{3}+\frac{5}{3}x.

$$z=\frac{200}{3}+\frac{5}{3}\left(50+0,5z\right)$$

Solve z=\frac{200}{3}+\frac{5}{3}\left(50+0,5z\right) for z.

$$z=900$$

Substitute 900 for z in the equation x=50+0,5z.

$$x=50+0,5\times 900$$

Calculate x from x=50+0,5\times 900.

$$x=500$$

Substitute 500 for x in the equation y=100+x.

$$y=100+500$$

Calculate y from y=100+500.

$$y=600$$

The system is now solved.

$$x=500 y=600 z=900$$