The formula for a direct variation is:

$\displaystyle \frac{x_1}{x_2}=\frac{y_1}{y_2}$

Plugging in our values, we get:

$\displaystyle \frac{7}{x}=\frac{21}{-5}$

$\displaystyle 21x=-35$

$\displaystyle x= \frac{-5}{3}$

The formula for an indirect variation is:

$\displaystyle \frac{x_1}{x_2}=\frac{y_2}{y_1}$

Plugging in our values, we get:

$\displaystyle \frac{60\ cm}{5\ cm}=\frac{y}{40\ cm}$

$\displaystyle 5y=2400\ cm$

$\displaystyle y=480\ cm$

This is a more complicated form of $\displaystyle \frac{1}{2}+\frac{1}{3}= \frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators.  Simplify as needed.

$\displaystyle \frac{x+2}{x-1}+\frac{x-5}{x+3}=\frac{x+2}{x-1}\cdot \frac{x+3}{x+3}+\frac{x-5}{x+3}\cdot \frac{x-1}{x-1}$

which is equivalent to $\displaystyle \frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}$

Simplify to get $\displaystyle \frac{2x^{2}-x+11}{x^{2}+2x-3}$

Multiply by the reciprocal of the second expression:

$\displaystyle \frac{x^2+7x+10}{x+2}\div \frac{x^2+2x-15}{x^2-5x+6}$

$\displaystyle \frac{x^2+7x+10}{x+2}\cdot \frac{x^2-5x+6}{x^2+2x-15}$

Factor the expressions:

$\displaystyle \frac{(x+5)(x+2)}{x+2}\cdot \frac{(x-3)(x-2)}{(x-3)(x+5)}$

Remove common terms:

$\displaystyle \frac{\mathbf{(x+5)(x+2)}}{\mathbf{x+2}}\cdot \frac{\mathbf{(x-3)}(x-2)}{\mathbf{(x-3)(x+5)}}$

$\displaystyle x-2$

Begin by multiplying the left term by $\displaystyle \frac{3n-1}{3n-1}$:

$\displaystyle 3n+1+\frac{1}{3n-1}$

$\displaystyle 3n+1 (\frac{3n-1}{3n-1})+\frac{1}{3n-1}$

Simplify:

$\displaystyle \frac{(3n+1)(3n-1)+1}{3n-1}$

$\displaystyle \frac{9n^2-1+1}{3n-1}$

$\displaystyle \frac{9n^2}{3n-1}$

Begin by combining the terms in the denominator:

$\displaystyle \frac{\frac{x+y}{x}}{\frac{1}{x}+\frac{1}{y}}$

$\displaystyle \frac{\frac{x+y}{x}}{\frac{x+y}{xy}}$

Multiply by the reciprocal of the denominator:

$\displaystyle \frac{x+y}{x} \cdot \frac{xy}{x+y}$

Remove like terms:

$\displaystyle \frac{\mathbf{x+y}}{\mathbf{x}} \cdot \frac{\boldsymbol{x}y}{\mathbf{x+y}}$

$\displaystyle y$

Create a common denominator of $\displaystyle (n+1)$ in both the numerator and denominator:

$\displaystyle \frac{n+5+\frac{3}{n+1}}{n-1-\frac{3}{n+1}}$

$\displaystyle \frac{\frac{(n+5)(n+1)+3}{n+1}}{\frac{(n-1)(n+1)-3}{n+1}}$

Multiply by the reciprocal of the denominator:

$\displaystyle \frac{(n+5)(n+1)+3}{n+1}\cdot \frac{n+1}{(n-1)(n+1)-3}$

Simplify:

$\displaystyle \frac{n^2+6n+8}{n+1}\cdot \frac{n+1}{n^2-4}$

$\displaystyle \frac{(n+4)(n+2)}{n+1}\cdot \frac{n+1}{(n-2)(n+2)}$

Remove common terms:

$\displaystyle \frac{(n+4)\mathbf{(n+2)}}{\mathbf{n+1}}\cdot \frac{\mathbf{n+1}}{(n-2)(\mathbf{n+2})}$

$\displaystyle \frac{n+4}{n-2}$

Factor the expression:

$\displaystyle \frac{(a-b)(a^2+ab+b^2)(a+b)}{(b-a)(b+a)(a^2+ab+b^2)}$

Remove like terms:

$\displaystyle \frac{(a-b)(\mathbf{a^2+ab+b^2})(\mathbf{a+b})}{(b-a)(\mathbf{b+a})(\mathbf{a^2+ab+b^2})}$

$\displaystyle \frac{a-b}{b-a}=\frac{-1(b-a)}{b-a}=-1$

Multiply by the inverse of the denominator:

$\displaystyle \frac{x^2-11x+24}{x^2-18x+80}\div \frac{x^2-9x+20}{x^2-15x+50}$

$\displaystyle \frac{x^2-11x+24}{x^2-18x+80}\cdot \frac{x^2-15x+50}{x^2-9x+20}$

Factor:

$\displaystyle \frac{(x-8)(x-3)}{(x-8)(x-10)}\cdot \frac{(x-5)(x-10)}{(x-5)(x-4)}$

Remove like terms:

$\displaystyle \frac{(\mathbf{x-8})(x-3)}{(\mathbf{x-8})(\mathbf{x-10})}\cdot \frac{(\mathbf{x-5})(\mathbf{x-10})}{(\mathbf{x-5})(x-4)}$

$\displaystyle \frac{x-3}{x-4}$

Multiply the equation by $\displaystyle (y-3)(y+3)$:

$\displaystyle \frac{y}{y-3}+\frac{6}{y+3}=1$

$\displaystyle y(y+3)+6(y-3)=1(y-3)(y+3)$

$\displaystyle y^2+3y+6y-18 = y^2-9$

$\displaystyle 9y=9$

$\displaystyle y=1$