Equations with Fractions

Solving an equation with fractions is the same as solving an equation without fractions.  The exact same rules apply: isolate the variables by using "opposite" operations to remove (or "undo") the terms that are attached to the variable.  "Undo" terms in reverse order of operations  (so, undo operations attached with addition and subtraction, then multiplication and division).

A fraction in an equation is just a division problem.  So if you have a term over a number, that translates into that term divided by that number.  To "undo" the division problem, muliply by the divisor.  Then procede with solving the equation.

Example:

$$\eqalign{\dfrac{4x}{3}&=12\qquad&&\text{Read: }4x\div3=12\\\times3&\;\;\times 3&&\text{Multiply each side by 3}\\4x&=36\\\div 4&\;\; \div 4 &&\text{Divide each side by 4}\\x&=9}$$

As you can see, the fraction just meant that $4x$ was dividied by 3. So, multiply each side of the equation by 3 to get rid of the denominator, then solve.

Sometimes you'll have equation with fractions on both sides (and no extra terms). In those cases, treat the equation like a proportion and cross-multiply to create an equation.  (Note: if there are extra addition or subtraction terms not in the fractions, you have to use the method above, cross-multiplying won't work.  But, if the extra terms are multiplication terms, you can just multiply them into the numerator of the adjacent fraction, see the example below).

Example:

$$\eqalign{\dfrac{2}{5}x&=\dfrac{3+x}{5}\qquad &&\text{Because }\dfrac{2}{5} \times x = \dfrac{2}{5}\times\dfrac{x}{1}=\dfrac{2x}{5}\text{, this is a proportion}\\\dfrac{2x}{5}&=\dfrac{3+x}{5}&&\text{So you can cross multiply and create an equation}\\2x\times5&=(3+x)\times5\\10x&=15 + 5x&&\text{Once you have an equation, just use algebra to solve for x}\\-5x&\qquad-5x\\5x&=15\\\div 5&=\;\;\div 5\\x&=3}$$

Fractions set equal to each other are proportions and most easily solved by cross multiplying.

Finally, sometimes you will get an equation with a variable in the denominator of a fraction. You want to clear variables out of denominators whenever possible.  How? You undo division by multiplying by the denominator, so multiply by the variable.  That will pull the variable up out of the denominator (and move it to the other side of the equal sign).  Then you can solve the way that you would solve any other equation. 

Example:

$$\eqalign{\dfrac{5}{x}&=12\qquad&&\text{Read: }5\div x=12\\\times x&\;\;\times x&&\text{Multiply each side by }x\\5&=12x\\\div 12&\;\; \div 12 &&\text{Divide each side by 12}\\x&=\dfrac{5}{12}}$$

Overall, don't let fractions in equations fool you -- it's just algebra!