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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:
$(\frac 2 4)= (\frac 1 2)$ . 
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:
$(\frac a b)=(\frac c d)$ 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 x; that is, find the value(s) that make the equation true.
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 x; this loses one of the two solutions.
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
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Key Term Reference
 denominator
 Appears in this related concepts: Finding the Domain of a Rational Function, Complex Conjugates and Division, and Rational Equations
 domain
 Appears in this related concepts: Finding Domains of Functions, VectorValued Functions, and Inverse Functions
 equation
 Appears in this related concepts: Motion with Constant Acceleration, A General Approach, and Equations and Inequalities
 expression
 Appears in this related concepts: Compound Inequalities, Expressions and Sets of Numbers, and Equations and Their Solutions
 factor
 Appears in this related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 fraction
 Appears in this related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 numerator
 Appears in this related concepts: Low Voter Turnout, Solving Problems with Rational Functions, and Rationalizing Denominators or Numerators
 quadratic
 Appears in this related concepts: Stretching and Shrinking, The Discriminant, and Quadratic Equations and Quadratic Functions
 quadratic equation
 Appears in this related concepts: Standard Form and Completing the Square, Completing the Square, and Linear and Quadratic Equations
 set
 Appears in this related concepts: Sequences of Statements, Sequences, and Introduction to Sequences
 variable
 Appears in this related concepts: Related Rates, Calculating the NPV, and Fundamentals of Statistics
Sources
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Cite This Source
Source: Boundless. “Solving Equations with Rational Expressions; Problems Involving Proportions.” Boundless Algebra. Boundless, 27 Jun. 2014. Retrieved 21 Mar. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/rationalexpressions13/solvingequationswithrationalexpressionsproblemsinvolvingproportions805540/