Equations with fractions

Summary and examples

Roughly speaking an equation with fractions is an equation with the independent variable (for example ) appearing in the denominator or in both the numerator and denominator.

In a number of examples we clarify how to solve equations with fractions.

Example 1

Solve:

The equation is only defined for denominators unequal to :

and

Essential in this first method is that fractions can only be added (or subtracted) if they have equal denominators. That is why we make both denominators equal:

resulting in the equation:

or:

or:

A fraction is equal to if the numerator is equal to , thus:

This equation is solved by applying the -formula:

A second method (cross-multiplication) is straightforward. Cross-multiplication means that the product of  and is made equal to the product and , thus:

resulting in the same equation.

Usually cross-multiplication is preferred because it is a faster method. However, this method fails if we have inequalities with fractions. We come back to this in detail in Inequalities with fractions.

Example 2

Solve:

The equation is only defined for denominators unequal to :

and

Cross-multiplication yields:

or:

so:

Example 3

Solve:

Both denominators are not equal to for all values of , and thus the equation is defined for alle values of .

Cross-multiplication yields:

This is a quadratic equation in and the solution is (using the -formula):

Just one of the two values is positive and thus valid since a power function cannot be negative. So:

and thus, after having applied the natural logarithm:

Example 4

Solve:

The equation is only valid if the denominators are unequal to :

en

In order to solve this equation we have to make the denominators in the left-hand side equal:

and thus:

0