Solve (91m-3n)^2-(91m+3n)^2=-108

$$( 91 m - 3 n ) ^ { 2 } - ( 91 m + 3 n ) ^ { 2 } = - 108$$

Solve for m

$$m=\frac{9}{91n}$$

$$n\neq 0$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(91m-3n\right)^{2}.

$$8281m^{2}-546mn+9n^{2}-\left(91m+3n\right)^{2}=-108$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(91m+3n\right)^{2}.

$$8281m^{2}-546mn+9n^{2}-\left(8281m^{2}+546mn+9n^{2}\right)=-108$$

To find the opposite of 8281m^{2}+546mn+9n^{2}, find the opposite of each term.

$$8281m^{2}-546mn+9n^{2}-8281m^{2}-546mn-9n^{2}=-108$$

Combine 8281m^{2} and -8281m^{2} to get 0.

$$-546mn+9n^{2}-546mn-9n^{2}=-108$$

Combine -546mn and -546mn to get -1092mn.

$$-1092mn+9n^{2}-9n^{2}=-108$$

Combine 9n^{2} and -9n^{2} to get 0.

$$-1092mn=-108$$

The equation is in standard form.

$$\left(-1092n\right)m=-108$$

Divide both sides by -1092n.

$$\frac{\left(-1092n\right)m}{-1092n}=-\frac{108}{-1092n}$$

Dividing by -1092n undoes the multiplication by -1092n.

$$m=-\frac{108}{-1092n}$$

Divide -108 by -1092n.

$$m=\frac{9}{91n}$$

Solve for n

$$n=\frac{9}{91m}$$

$$m\neq 0$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(91m-3n\right)^{2}.

$$8281m^{2}-546mn+9n^{2}-\left(91m+3n\right)^{2}=-108$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(91m+3n\right)^{2}.

$$8281m^{2}-546mn+9n^{2}-\left(8281m^{2}+546mn+9n^{2}\right)=-108$$

To find the opposite of 8281m^{2}+546mn+9n^{2}, find the opposite of each term.

$$8281m^{2}-546mn+9n^{2}-8281m^{2}-546mn-9n^{2}=-108$$

Combine 8281m^{2} and -8281m^{2} to get 0.

$$-546mn+9n^{2}-546mn-9n^{2}=-108$$

Combine -546mn and -546mn to get -1092mn.

$$-1092mn+9n^{2}-9n^{2}=-108$$

Combine 9n^{2} and -9n^{2} to get 0.

$$-1092mn=-108$$

The equation is in standard form.

$$\left(-1092m\right)n=-108$$

Divide both sides by -1092m.

$$\frac{\left(-1092m\right)n}{-1092m}=-\frac{108}{-1092m}$$

Dividing by -1092m undoes the multiplication by -1092m.

$$n=-\frac{108}{-1092m}$$

Divide -108 by -1092m.

$$n=\frac{9}{91m}$$