Solve {l}{3/4x+5/2y=8}{0.25y-0.4z=0.1}{-x/2-z/2=-frac{5}{2}}

$$\left\{ \begin{array} { l } { \frac { 3 } { 4 } x + \frac { 5 } { 2 } y = 8 } \\ { 0.25 y - 0.4 z = 0.1 } \\ { - \frac { x } { 2 } - \frac { z } { 2 } = - \frac { 5 } { 2 } } \end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

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

$$3x+10y=32 0.25y-0.4z=0.1 -x-z=-5$$

Reorder the equations.

$$-x-z=-5 0.25y-0.4z=0.1 3x+10y=32$$

Solve -x-z=-5 for x.

$$x=-z+5$$

Substitute -z+5 for x in the equation 3x+10y=32.

$$3\left(-z+5\right)+10y=32$$

Solve the second equation for y and the third equation for z.

$$y=1.6z+0.4 z=-\frac{17}{3}+\frac{10}{3}y$$

Substitute 1.6z+0.4 for y in the equation z=-\frac{17}{3}+\frac{10}{3}y.

$$z=-\frac{17}{3}+\frac{10}{3}\left(1.6z+0.4\right)$$

Solve z=-\frac{17}{3}+\frac{10}{3}\left(1.6z+0.4\right) for z.

$$z=1$$

Substitute 1 for z in the equation y=1.6z+0.4.

$$y=1.6\times 1+0.4$$

Calculate y from y=1.6\times 1+0.4.

$$y=2$$

Substitute 1 for z in the equation x=-z+5.

$$x=-1+5$$

Calculate x from x=-1+5.

$$x=4$$

The system is now solved.

$$x=4 y=2 z=1$$