Solve {l}{x+y+z=2/8}{3x+4y+z=9}{5x+2y+5z=3}

$$\left. \begin{array} { l } { x + y + z = \frac { 2 } { 8 } } \\ { 3 x + 4 y + z = 9 } \\ { 5 x + 2 y + 5 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{21}{4} = 5\frac{1}{4} = 5.25$$

$$y=-\frac{7}{12}\approx -0.583333333$$

$$z = -\frac{53}{12} = -4\frac{5}{12} \approx -4.416666667$$

Short Steps Using Substitution

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

$$8x+8y+8z=2 3x+4y+z=9 5x+2y+5z=3$$

Reorder the equations.

$$3x+4y+z=9 8x+8y+8z=2 5x+2y+5z=3$$

Solve 3x+4y+z=9 for z.

$$z=-3x-4y+9$$

Substitute -3x-4y+9 for z in the second and third equation.

$$8x+8y+8\left(-3x-4y+9\right)=2 5x+2y+5\left(-3x-4y+9\right)=3$$

Solve these equations for y and x respectively.

$$y=\frac{35}{12}-\frac{2}{3}x x=\frac{21}{5}-\frac{9}{5}y$$

Substitute \frac{35}{12}-\frac{2}{3}x for y in the equation x=\frac{21}{5}-\frac{9}{5}y.

$$x=\frac{21}{5}-\frac{9}{5}\left(\frac{35}{12}-\frac{2}{3}x\right)$$

Solve x=\frac{21}{5}-\frac{9}{5}\left(\frac{35}{12}-\frac{2}{3}x\right) for x.

$$x=\frac{21}{4}$$

Substitute \frac{21}{4} for x in the equation y=\frac{35}{12}-\frac{2}{3}x.

$$y=\frac{35}{12}-\frac{2}{3}\times \frac{21}{4}$$

Calculate y from y=\frac{35}{12}-\frac{2}{3}\times \frac{21}{4}.

$$y=-\frac{7}{12}$$

Substitute -\frac{7}{12} for y and \frac{21}{4} for x in the equation z=-3x-4y+9.

$$z=-3\times \frac{21}{4}-4\left(-\frac{7}{12}\right)+9$$

Calculate z from z=-3\times \frac{21}{4}-4\left(-\frac{7}{12}\right)+9.

$$z=-\frac{53}{12}$$

The system is now solved.

$$x=\frac{21}{4} y=-\frac{7}{12} z=-\frac{53}{12}$$