Solve ax+b/cx+d=A

$$\frac { a x + b } { c x + d } = A$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{Acx+Ad-b}{x}\text{, }&x\neq 0\text{ and }d\neq -cx\\a\in \mathrm{C}\text{, }&b=Ad\text{ and }x=0\text{ and }d\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by cx+d.

$$ax+b=A\left(cx+d\right)$$

Use the distributive property to multiply A by cx+d.

$$ax+b=Acx+Ad$$

Subtract b from both sides.

$$ax=Acx+Ad-b$$

The equation is in standard form.

$$xa=Acx+Ad-b$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{Acx+Ad-b}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{Acx+Ad-b}{x}$$

Solve for A

$$A=\frac{ax+b}{cx+d}$$

$$d\neq -cx$$

Steps for Solving Linear Equation

Multiply both sides of the equation by cx+d.

$$ax+b=A\left(cx+d\right)$$

Use the distributive property to multiply A by cx+d.

$$ax+b=Acx+Ad$$

Swap sides so that all variable terms are on the left hand side.

$$Acx+Ad=ax+b$$

Combine all terms containing A.

$$\left(cx+d\right)A=ax+b$$

Divide both sides by cx+d.

$$\frac{\left(cx+d\right)A}{cx+d}=\frac{ax+b}{cx+d}$$

Dividing by cx+d undoes the multiplication by cx+d.

$$A=\frac{ax+b}{cx+d}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{Acx+Ad-b}{x}\text{, }&x\neq 0\text{ and }d\neq -cx\\a\in \mathrm{R}\text{, }&b=Ad\text{ and }x=0\text{ and }d\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by cx+d.

$$ax+b=A\left(cx+d\right)$$

Use the distributive property to multiply A by cx+d.

$$ax+b=Acx+Ad$$

Subtract b from both sides.

$$ax=Acx+Ad-b$$

The equation is in standard form.

$$xa=Acx+Ad-b$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{Acx+Ad-b}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{Acx+Ad-b}{x}$$