Solve x/2+y/3+z/4=x+y+z/5

$$\frac { x } { 2 } + \frac { y } { 3 } + \frac { z } { 4 } = \frac { x + y + z } { 5 }$$

Solve for x

$$x=-\frac{z}{6}-\frac{4y}{9}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 60, the least common multiple of 2,3,4,5.

$$30x+20y+15z=12\left(x+y+z\right)$$

Use the distributive property to multiply 12 by x+y+z.

$$30x+20y+15z=12x+12y+12z$$

Subtract 12x from both sides.

$$30x+20y+15z-12x=12y+12z$$

Combine 30x and -12x to get 18x.

$$18x+20y+15z=12y+12z$$

Subtract 20y from both sides.

$$18x+15z=12y+12z-20y$$

Combine 12y and -20y to get -8y.

$$18x+15z=-8y+12z$$

Subtract 15z from both sides.

$$18x=-8y+12z-15z$$

Combine 12z and -15z to get -3z.

$$18x=-8y-3z$$

Divide both sides by 18.

$$\frac{18x}{18}=\frac{-8y-3z}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x=\frac{-8y-3z}{18}$$

Divide -8y-3z by 18.

$$x=-\frac{z}{6}-\frac{4y}{9}$$

Solve for y

$$y=-\frac{3z}{8}-\frac{9x}{4}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 60, the least common multiple of 2,3,4,5.

$$30x+20y+15z=12\left(x+y+z\right)$$

Use the distributive property to multiply 12 by x+y+z.

$$30x+20y+15z=12x+12y+12z$$

Subtract 12y from both sides.

$$30x+20y+15z-12y=12x+12z$$

Combine 20y and -12y to get 8y.

$$30x+8y+15z=12x+12z$$

Subtract 30x from both sides.

$$8y+15z=12x+12z-30x$$

Combine 12x and -30x to get -18x.

$$8y+15z=-18x+12z$$

Subtract 15z from both sides.

$$8y=-18x+12z-15z$$

Combine 12z and -15z to get -3z.

$$8y=-18x-3z$$

Divide both sides by 8.

$$\frac{8y}{8}=\frac{-18x-3z}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$y=\frac{-18x-3z}{8}$$

Divide -18x-3z by 8.

$$y=-\frac{3z}{8}-\frac{9x}{4}$$