Solve ay+b=3b+by

$$a y + b = 3 b + b y$$

Solve for a

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

Steps for Solving Linear Equation

Subtract b from both sides.

$$ay=3b+by-b$$

Combine 3b and -b to get 2b.

$$ay=2b+by$$

The equation is in standard form.

$$ya=by+2b$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{b\left(y+2\right)}{y}$$

Dividing by y undoes the multiplication by y.

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

Solve for b

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

Steps for Solving Linear Equation

Subtract 3b from both sides.

$$ay+b-3b=by$$

Combine b and -3b to get -2b.

$$ay-2b=by$$

Subtract by from both sides.

$$ay-2b-by=0$$

Subtract ay from both sides. Anything subtracted from zero gives its negation.

$$-2b-by=-ay$$

Reorder the terms.

$$-by-2b=-ay$$

Combine all terms containing b.

$$\left(-y-2\right)b=-ay$$

Divide both sides by -2-y.

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

Dividing by -2-y undoes the multiplication by -2-y.

$$b=-\frac{ay}{-y-2}$$

Divide -ay by -2-y.

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