Solve 18mn+42m+6n+14=(6m+A)(3n+B)

$$18 m n + 42 m + 6 n + 14 = ( 6 m + A ) ( 3 n + B )$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=-\frac{2\left(3Bm-21m-3n-7\right)}{3n+B}\text{, }&n\neq -\frac{B}{3}\\A\in \mathrm{C}\text{, }&\left(m=-\frac{1}{3}\text{ and }n=-\frac{B}{3}\right)\text{ or }\left(n=-\frac{7}{3}\text{ and }B=7\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 6m+A by 3n+B.

$$18mn+42m+6n+14=18mn+6mB+3An+AB$$

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

$$18mn+6mB+3An+AB=18mn+42m+6n+14$$

Subtract 18mn from both sides.

$$6mB+3An+AB=18mn+42m+6n+14-18mn$$

Combine 18mn and -18mn to get 0.

$$6mB+3An+AB=42m+6n+14$$

Subtract 6mB from both sides.

$$3An+AB=42m+6n+14-6mB$$

Combine all terms containing A.

$$\left(3n+B\right)A=42m+6n+14-6mB$$

The equation is in standard form.

$$\left(3n+B\right)A=14+6n+42m-6Bm$$

Divide both sides by 3n+B.

$$\frac{\left(3n+B\right)A}{3n+B}=\frac{14+6n+42m-6Bm}{3n+B}$$

Dividing by 3n+B undoes the multiplication by 3n+B.

$$A=\frac{14+6n+42m-6Bm}{3n+B}$$

Divide 42m+6n+14-6mB by 3n+B.

$$A=\frac{2\left(7+3n+21m-3Bm\right)}{3n+B}$$

Solve for B (complex solution)

$$\left\{\begin{matrix}B=-\frac{-42m+3An-6n-14}{6m+A}\text{, }&m\neq -\frac{A}{6}\\B\in \mathrm{C}\text{, }&\left(m=-\frac{1}{3}\text{ and }A=2\right)\text{ or }\left(n=-\frac{7}{3}\text{ and }m=-\frac{A}{6}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 6m+A by 3n+B.

$$18mn+42m+6n+14=18mn+6mB+3An+AB$$

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

$$18mn+6mB+3An+AB=18mn+42m+6n+14$$

Subtract 18mn from both sides.

$$6mB+3An+AB=18mn+42m+6n+14-18mn$$

Combine 18mn and -18mn to get 0.

$$6mB+3An+AB=42m+6n+14$$

Subtract 3An from both sides.

$$6mB+AB=42m+6n+14-3An$$

Combine all terms containing B.

$$\left(6m+A\right)B=42m+6n+14-3An$$

The equation is in standard form.

$$\left(6m+A\right)B=42m-3An+6n+14$$

Divide both sides by 6m+A.

$$\frac{\left(6m+A\right)B}{6m+A}=\frac{42m-3An+6n+14}{6m+A}$$

Dividing by 6m+A undoes the multiplication by 6m+A.

$$B=\frac{42m-3An+6n+14}{6m+A}$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{2\left(3Bm-21m-3n-7\right)}{3n+B}\text{, }&n\neq -\frac{B}{3}\\A\in \mathrm{R}\text{, }&\left(m=-\frac{1}{3}\text{ and }n=-\frac{B}{3}\right)\text{ or }\left(n=-\frac{7}{3}\text{ and }B=7\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 6m+A by 3n+B.

$$18mn+42m+6n+14=18mn+6mB+3An+AB$$

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

$$18mn+6mB+3An+AB=18mn+42m+6n+14$$

Subtract 18mn from both sides.

$$6mB+3An+AB=18mn+42m+6n+14-18mn$$

Combine 18mn and -18mn to get 0.

$$6mB+3An+AB=42m+6n+14$$

Subtract 6mB from both sides.

$$3An+AB=42m+6n+14-6mB$$

Combine all terms containing A.

$$\left(3n+B\right)A=42m+6n+14-6mB$$

The equation is in standard form.

$$\left(3n+B\right)A=14+6n+42m-6Bm$$

Divide both sides by 3n+B.

$$\frac{\left(3n+B\right)A}{3n+B}=\frac{14+6n+42m-6Bm}{3n+B}$$

Dividing by 3n+B undoes the multiplication by 3n+B.

$$A=\frac{14+6n+42m-6Bm}{3n+B}$$

Divide 42m+6n+14-6mB by 3n+B.

$$A=\frac{2\left(7+3n+21m-3Bm\right)}{3n+B}$$

Solve for B

$$\left\{\begin{matrix}B=-\frac{-42m+3An-6n-14}{6m+A}\text{, }&m\neq -\frac{A}{6}\\B\in \mathrm{R}\text{, }&\left(m=-\frac{1}{3}\text{ and }A=2\right)\text{ or }\left(n=-\frac{7}{3}\text{ and }m=-\frac{A}{6}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 6m+A by 3n+B.

$$18mn+42m+6n+14=18mn+6mB+3An+AB$$

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

$$18mn+6mB+3An+AB=18mn+42m+6n+14$$

Subtract 18mn from both sides.

$$6mB+3An+AB=18mn+42m+6n+14-18mn$$

Combine 18mn and -18mn to get 0.

$$6mB+3An+AB=42m+6n+14$$

Subtract 3An from both sides.

$$6mB+AB=42m+6n+14-3An$$

Combine all terms containing B.

$$\left(6m+A\right)B=42m+6n+14-3An$$

The equation is in standard form.

$$\left(6m+A\right)B=42m-3An+6n+14$$

Divide both sides by 6m+A.

$$\frac{\left(6m+A\right)B}{6m+A}=\frac{42m-3An+6n+14}{6m+A}$$

Dividing by 6m+A undoes the multiplication by 6m+A.

$$B=\frac{42m-3An+6n+14}{6m+A}$$