Solve {l}{x/3+y/2-z/5=9}{x-2y+z=1}{x+y/3=z-1}

$$\left\{ \begin{array} { l } { \frac { x } { 3 } + \frac { y } { 2 } - \frac { z } { 5 } = 9 } \\ { x - 2 y + z = 1 } \\ { \frac { x + y } { 3 } = z - 1 } \end{array} \right.$$

Solve for x, y, z

$$x=15$$

$$y=12$$

$$z=10$$

Short Steps Using Substitution

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

$$10x+15y-6z=270 x-2y+z=1 x+y=3z-3$$

Reorder the equations.

$$x-2y+z=1 10x+15y-6z=270 x+y=3z-3$$

Solve x-2y+z=1 for x.

$$x=2y-z+1$$

Substitute 2y-z+1 for x in the second and third equation.

$$10\left(2y-z+1\right)+15y-6z=270 2y-z+1+y=3z-3$$

Solve these equations for y and z respectively.

$$y=\frac{52}{7}+\frac{16}{35}z z=\frac{3}{4}y+1$$

Substitute \frac{52}{7}+\frac{16}{35}z for y in the equation z=\frac{3}{4}y+1.

$$z=\frac{3}{4}\left(\frac{52}{7}+\frac{16}{35}z\right)+1$$

Solve z=\frac{3}{4}\left(\frac{52}{7}+\frac{16}{35}z\right)+1 for z.

$$z=10$$

Substitute 10 for z in the equation y=\frac{52}{7}+\frac{16}{35}z.

$$y=\frac{52}{7}+\frac{16}{35}\times 10$$

Calculate y from y=\frac{52}{7}+\frac{16}{35}\times 10.

$$y=12$$

Substitute 12 for y and 10 for z in the equation x=2y-z+1.

$$x=2\times 12-10+1$$

Calculate x from x=2\times 12-10+1.

$$x=15$$

The system is now solved.

$$x=15 y=12 z=10$$