Solve a/x+b/y=a/2+b/3

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

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

$$6ya+6xb=3xya+2xyb$$

Subtract 3xya from both sides.

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

Subtract 6xb from both sides.

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

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by -3xy+6y.

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

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

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

Divide 2xb\left(-3+y\right) by -3xy+6y.

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

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

$$6ya+6xb=3xya+2xyb$$

Subtract 2xyb from both sides.

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

Subtract 6ya from both sides.

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

Combine all terms containing b.

$$\left(6x-2xy\right)b=3xya-6ya$$

The equation is in standard form.

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

Divide both sides by 6x-2yx.

$$\frac{\left(6x-2xy\right)b}{6x-2xy}=\frac{3ay\left(x-2\right)}{6x-2xy}$$

Dividing by 6x-2yx undoes the multiplication by 6x-2yx.

$$b=\frac{3ay\left(x-2\right)}{6x-2xy}$$

Divide 3ay\left(-2+x\right) by 6x-2yx.

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

Solve for a

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

Steps for Solving Linear Equation

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

$$6ya+6xb=3xya+2xyb$$

Subtract 3xya from both sides.

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

Subtract 6xb from both sides.

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

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by -3xy+6y.

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

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

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

Divide 2xb\left(-3+y\right) by -3xy+6y.

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

Solve for b

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

Steps for Solving Linear Equation

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

$$6ya+6xb=3xya+2xyb$$

Subtract 2xyb from both sides.

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

Subtract 6ya from both sides.

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

Combine all terms containing b.

$$\left(6x-2xy\right)b=3xya-6ya$$

The equation is in standard form.

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

Divide both sides by 6x-2yx.

$$\frac{\left(6x-2xy\right)b}{6x-2xy}=\frac{3ay\left(x-2\right)}{6x-2xy}$$

Dividing by 6x-2yx undoes the multiplication by 6x-2yx.

$$b=\frac{3ay\left(x-2\right)}{6x-2xy}$$

Divide 3ay\left(-2+x\right) by 6x-2yx.

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