Solving Rational Equations
Solving rational equations involves finding the values of variables in equations that are expressed as the ratio of two polynomials. A rational equation equates one rational expression to another, where a rational expression consists of integer constants, variables, and basic algebraic operations. The simplest method to solve these equations is by multiplying each side to achieve a common denominator, thereby eliminating denominators and simplifying the equation. While some rational equations can be solved easily, others may require more complex manipulation.
In cases where the variable appears in the denominator, caution is necessary, as solutions may inadvertently lead to division by zero, indicating that the equation has no valid solution. Additionally, solutions may yield multiple answers, particularly when exponents are involved, necessitating verification by substituting back into the original equation. Graphing can be a helpful tool in visualizing the solutions and confirming their validity. Understanding these concepts is essential for effectively navigating and solving rational equations in algebra.
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Subject Terms
Solving Rational Equations
In algebra, a rational equation is one in which one rational expression is defined as equal to another. A rational expression is any algebraic expression (made up of integer constants, variables, and the algebraic operations of addition, subtraction, multiplication, division, and exponentiation) that may be constructed as a rational fraction. Some rational equations solve very simply, as in 4/75 = x/75. Since the denominators are the same, x = 4.
Others require more work in order to convert each side of the equation to a common denominator. It is important to remember that rational equations do not include all algebraic equations. Most notably, square roots and other roots are not used in rational expressions. Further, algebraic expressions themselves are more constrained in their contents than mathematical expressions as a whole, for example, lacking irrational exponents, trigonometric functions like sine and tangent, logarithms, the functions (gamma, bessel, special, or continued) and infinite series of analytical expressions, and the limits, integrals, and differentials of other mathematical expressions.
Overview
Multiplying each side of a rational equation in order to convert to a common denominator is the simplest way to solve a rational equation. While 2/3 = x/12 is not difficult to solve off the top of one's head (8 is two-thirds of 12, so x = 8), 52/933 = x/478 is much trickier. Multiplying 933 by 478 results in a common denominator of 445,974. The numerator of each side is multiplied by the denominator of the other:
(52 × 478)/445,974 = (933x)/445,974.
Therefore 52 × 478 = 933x, or 24,856 = 933x, or 24856/933 = x
Which means x = 24.6409
This method is simplified as "the product of the means equals the product of the extremes." For a rational equation a/b = c/d, ad = bc.
While the denominator has been eliminated here, for accuracy, it needs to be checked against the original equation. In this case, there is no problem—24.6409/478 is a clunky number, but a valid one. Had the answer resulted in turning the denominator to zero or otherwise required dividing by zero, this would mean that, even though we had appeared to "solve" it, the rational equation actually had no answer. This is particularly a risk when the variable x is in the denominator rather than the numerator, as in the example.) Similarly, some solutions to rational equations will result in two answers—one positive, one negative—because of an exponent in the equation. These answers must be plugged back into the original equation to confirm that their results are valid. In some cases, only one of the two answers will prove to be valid. Graphing the equation can help with this.
Bibliography
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