Solving Equations with Rational Expressions; Problems Involving Proportions
Rational expressions, like proportions, are extremely useful applications of algebra, that can be solved using simple algebraic techniques.
Learning Objective

Solve equations with rational expressions (proportions) by finding the LCD or by crossmultipliation
Key Points
 A rational equation means that you are setting two rational expressions equal to each other. Proportions are perfect examples of a rational expression. Even if they look different, they can be simplified down into the same expression:
$\displaystyle \frac 2 4= \frac 1 2$ .  If you have a rational equation where the denominators are the same, then the numerators must be the same. This in turn suggests a strategy: find a common denominator, and then set the numerators equal using algebraic techniques.
 Remember, all normal algebraic rules apply to solving rational equations. Such as, you still can not divide by 0.
Term

rational expression
An expression that can be expressed as the quotient of two polynomials.
Example
 When given the rational equation:
$\displaystyle\frac a b=\frac c d$ This can be solved by either finding a common denominator, or by setting it up like:$ad=cb$ and then solving it algebraically.
Full Text
A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for
Suppose you are told that:
If you think about it, the x in this equation has to be a 3. That is to say, if
This leads us to a very general rule: If you have a rational equation where the denominators are the same, then the numerators must be the same.
This in turn suggests a strategy: find a common denominator, and then set the numerators equal.
For example, consider the rational equation
by factoring the denominators,we find that we must multiply the left side of the equation by
Based on the rule above—since the denominators are equal, we can now assume the numerators are equal, so we know that
What we're dealing with, in this case, is a quadratic equation. As always, move everything to one side, giving
and then factor. A common mistake in this kind of problem is to divide both sides by
Two solutions to the quadratic equation. However, in this case,
As always, it is vital to remember what we have found here. We started with the equation
Key Term Reference
 denominator
 Appears in these related concepts: Introduction to Rational Functions, Solving Problems with Rational Functions, and Fractions Involving Radicals
 domain
 Appears in these related concepts: Visualizing Domain and Range, Restricting Domains to Find Inverses, and Composition of Functions and Decomposing a Function
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 factor
 Appears in these related concepts: Rational Algebraic Expressions, Factors, and Finding Factors of Polynomials
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 numerator
 Appears in these related concepts: Low Voter Turnout, Rational Equations, and Permutations of Distinguishable Objects
 quadratic
 Appears in these related concepts: The Discriminant, Stretching and Shrinking, and What is a Quadratic Function?
 quadratic equation
 Appears in these related concepts: Graphing Quadratic Equations In Standard Form, Linear and Quadratic Equations, and Completing the Square
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources: