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

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

Solve for x, z, y

$$x = \frac{136}{21} = 6\frac{10}{21} \approx 6.476190476$$

$$y = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

$$z = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

Short Steps Using Substitution

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

$$15x+16z=120 15x+8y=120 2y+3z=10$$

Solve 15x+16z=120 for x.

$$x=8-\frac{16}{15}z$$

Substitute 8-\frac{16}{15}z for x in the equation 15x+8y=120.

$$15\left(8-\frac{16}{15}z\right)+8y=120$$

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

$$z=\frac{1}{2}y y=-\frac{3}{2}z+5$$

Substitute \frac{1}{2}y for z in the equation y=-\frac{3}{2}z+5.

$$y=-\frac{3}{2}\times \frac{1}{2}y+5$$

Solve y=-\frac{3}{2}\times \frac{1}{2}y+5 for y.

$$y=\frac{20}{7}$$

Substitute \frac{20}{7} for y in the equation z=\frac{1}{2}y.

$$z=\frac{1}{2}\times \frac{20}{7}$$

Calculate z from z=\frac{1}{2}\times \frac{20}{7}.

$$z=\frac{10}{7}$$

Substitute \frac{10}{7} for z in the equation x=8-\frac{16}{15}z.

$$x=8-\frac{16}{15}\times \frac{10}{7}$$

Calculate x from x=8-\frac{16}{15}\times \frac{10}{7}.

$$x=\frac{136}{21}$$

The system is now solved.

$$x=\frac{136}{21} z=\frac{10}{7} y=\frac{20}{7}$$