Solve a/x-b/y=a/2-1/3-a

$$\frac { a } { x } - \frac { b } { y } = \frac { a } { 2 } - \frac { 1 } { 3 } - a$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{2x\left(y-3b\right)}{3y\left(x+2\right)}\text{, }&x\neq -2\text{ and }y\neq 0\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&x=-2\text{ and }y\neq 0\text{ and }b=\frac{y}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6xy, the least common multiple of x,y,2,3.

$$6ya-6xb=3xya+6xy\left(-\frac{1}{3}\right)-a\times 6xy$$

Multiply 6 and -\frac{1}{3} to get -2.

$$6ya-6xb=3xya-2xy-a\times 6xy$$

Multiply -1 and 6 to get -6.

$$6ya-6xb=3xya-2xy-6axy$$

Combine 3xya and -6axy to get -3xya.

$$6ya-6xb=-3xya-2xy$$

Add 3xya to both sides.

$$6ya-6xb+3xya=-2xy$$

Add 6xb to both sides.

$$6ya+3xya=-2xy+6xb$$

Combine all terms containing a.

$$\left(6y+3xy\right)a=-2xy+6xb$$

The equation is in standard form.

$$\left(3xy+6y\right)a=6bx-2xy$$

Divide both sides by 3xy+6y.

$$\frac{\left(3xy+6y\right)a}{3xy+6y}=\frac{6bx-2xy}{3xy+6y}$$

Dividing by 3xy+6y undoes the multiplication by 3xy+6y.

$$a=\frac{6bx-2xy}{3xy+6y}$$

Divide -2xy+6xb by 3xy+6y.

$$a=\frac{2x\left(3b-y\right)}{3y\left(x+2\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{2x\left(y-3b\right)}{3y\left(x+2\right)}\text{, }&x\neq -2\text{ and }y\neq 0\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&x=-2\text{ and }y\neq 0\text{ and }b=\frac{y}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6xy, the least common multiple of x,y,2,3.

$$6ya-6xb=3xya+6xy\left(-\frac{1}{3}\right)-a\times 6xy$$

Multiply 6 and -\frac{1}{3} to get -2.

$$6ya-6xb=3xya-2xy-a\times 6xy$$

Multiply -1 and 6 to get -6.

$$6ya-6xb=3xya-2xy-6axy$$

Combine 3xya and -6axy to get -3xya.

$$6ya-6xb=-3xya-2xy$$

Add 3xya to both sides.

$$6ya-6xb+3xya=-2xy$$

Add 6xb to both sides.

$$6ya+3xya=-2xy+6xb$$

Combine all terms containing a.

$$\left(6y+3xy\right)a=-2xy+6xb$$

The equation is in standard form.

$$\left(3xy+6y\right)a=6bx-2xy$$

Divide both sides by 3xy+6y.

$$\frac{\left(3xy+6y\right)a}{3xy+6y}=\frac{6bx-2xy}{3xy+6y}$$

Dividing by 3xy+6y undoes the multiplication by 3xy+6y.

$$a=\frac{6bx-2xy}{3xy+6y}$$

Divide -2xy+6xb by 3xy+6y.

$$a=\frac{2x\left(3b-y\right)}{3y\left(x+2\right)}$$

Solve for b

$$b=\frac{ay}{2}+\frac{ay}{x}+\frac{y}{3}$$

$$x\neq 0\text{ and }y\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6xy, the least common multiple of x,y,2,3.

$$6ya-6xb=3xya+6xy\left(-\frac{1}{3}\right)-a\times 6xy$$

Multiply 6 and -\frac{1}{3} to get -2.

$$6ya-6xb=3xya-2xy-a\times 6xy$$

Multiply -1 and 6 to get -6.

$$6ya-6xb=3xya-2xy-6axy$$

Combine 3xya and -6axy to get -3xya.

$$6ya-6xb=-3xya-2xy$$

Subtract 6ya from both sides.

$$-6xb=-3xya-2xy-6ya$$

The equation is in standard form.

$$\left(-6x\right)b=-3axy-2xy-6ay$$

Divide both sides by -6x.

$$\frac{\left(-6x\right)b}{-6x}=-\frac{y\left(3ax+2x+6a\right)}{-6x}$$

Dividing by -6x undoes the multiplication by -6x.

$$b=-\frac{y\left(3ax+2x+6a\right)}{-6x}$$

Divide -y\left(3xa+2x+6a\right) by -6x.

$$b=\frac{ay}{2}+\frac{ay}{x}+\frac{y}{3}$$