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

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

Solve for x, y, z

$$x=-38,8$$

$$y=61,2$$

$$z=-177,6$$

Short Steps Using Substitution

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

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

Reorder the equations.

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

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

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

Solve these equations for z and x respectively.

$$z=-100+2x x=-38,8$$

Substitute -38,8 for x in the equation z=-100+2x.

$$z=-100+2\left(-38,8\right)$$

Calculate z from z=-100+2\left(-38,8\right).

$$z=-177,6$$

Substitute -38,8 for x in the equation y=100+x.

$$y=100-38,8$$

Calculate y from y=100-38,8.

$$y=61,2$$

The system is now solved.

$$x=-38,8 y=61,2 z=-177,6$$