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

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

Solve for x

$$x=\frac{3y}{2}-\frac{3y}{4z}+\frac{15}{8}$$

$$z\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6z, the least common multiple of z,3,2.

$$2x\times 6z+6y=\frac{2}{3}x\times 6z+2y\times 6z+6z\times \frac{5}{2}$$

Multiply 2 and 6 to get 12.

$$12xz+6y=\frac{2}{3}x\times 6z+2y\times 6z+6z\times \frac{5}{2}$$

Multiply \frac{2}{3} and 6 to get 4.

$$12xz+6y=4xz+2y\times 6z+6z\times \frac{5}{2}$$

Multiply 2 and 6 to get 12.

$$12xz+6y=4xz+12yz+6z\times \frac{5}{2}$$

Multiply 6 and \frac{5}{2} to get 15.

$$12xz+6y=4xz+12yz+15z$$

Subtract 4xz from both sides.

$$12xz+6y-4xz=12yz+15z$$

Combine 12xz and -4xz to get 8xz.

$$8xz+6y=12yz+15z$$

Subtract 6y from both sides.

$$8xz=12yz+15z-6y$$

The equation is in standard form.

$$8zx=12yz-6y+15z$$

Divide both sides by 8z.

$$\frac{8zx}{8z}=\frac{12yz-6y+15z}{8z}$$

Dividing by 8z undoes the multiplication by 8z.

$$x=\frac{12yz-6y+15z}{8z}$$

Divide 12yz+15z-6y by 8z.

$$x=\frac{3y}{2}-\frac{3y}{4z}+\frac{15}{8}$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{z\left(8x-15\right)}{6\left(1-2z\right)}\text{, }&z\neq \frac{1}{2}\text{ and }z\neq 0\\y\in \mathrm{R}\text{, }&z=\frac{1}{2}\text{ and }x=\frac{15}{8}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6z, the least common multiple of z,3,2.

$$2x\times 6z+6y=\frac{2}{3}x\times 6z+2y\times 6z+6z\times \frac{5}{2}$$

Multiply 2 and 6 to get 12.

$$12xz+6y=\frac{2}{3}x\times 6z+2y\times 6z+6z\times \frac{5}{2}$$

Multiply \frac{2}{3} and 6 to get 4.

$$12xz+6y=4xz+2y\times 6z+6z\times \frac{5}{2}$$

Multiply 2 and 6 to get 12.

$$12xz+6y=4xz+12yz+6z\times \frac{5}{2}$$

Multiply 6 and \frac{5}{2} to get 15.

$$12xz+6y=4xz+12yz+15z$$

Subtract 12yz from both sides.

$$12xz+6y-12yz=4xz+15z$$

Subtract 12xz from both sides.

$$6y-12yz=4xz+15z-12xz$$

Combine 4xz and -12xz to get -8xz.

$$6y-12yz=-8xz+15z$$

Combine all terms containing y.

$$\left(6-12z\right)y=-8xz+15z$$

The equation is in standard form.

$$\left(6-12z\right)y=15z-8xz$$

Divide both sides by 6-12z.

$$\frac{\left(6-12z\right)y}{6-12z}=\frac{z\left(15-8x\right)}{6-12z}$$

Dividing by 6-12z undoes the multiplication by 6-12z.

$$y=\frac{z\left(15-8x\right)}{6-12z}$$

Divide z\left(15-8x\right) by 6-12z.

$$y=\frac{z\left(15-8x\right)}{6\left(1-2z\right)}$$