How to Add and Subtract Algebraic Fractions
In arithmetic, adding and subtracting fractions is a relatively straightforward process. Take 1/2 + 1/3. The easiest way to perform the operation is to multiply denominators 2 times 3 and use 6 as the common denominator: 1 times 3 is 3, and 2 times 3 is 6, so the first fraction is 3/6. And 1 times 2 is 2, and 3 times 2 is 6, so the second fraction is 2/6. Adding the two fractions 3/6 + 2/6 gives us 5/6. An algebraic fraction is a fraction with at least one variable in the denominator. The addition and subtraction of algebraic fractions follows procedures similar to those used for regular fractions.
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Instructions
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Take a relatively easy problem: x/2 + x/3. This problem asks you to add algebraic fractions. Since 2 times 3 is 6, we can use 6 as a common denominator.
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Convert the fractions to the common denominator 6. x/2 = 3x/6 because x times 3 is 3x and 2 times 3 is 6. x/3 = 2x/6 because x times 2 is 2x and 3 times 2 is 6.
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Add the new fractions: 3x/6 + 2x/6 = 5x/6. This is the final answer.
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Recognize that some problems may require you to factor algebraic expressions. Take an example: x/(x+y) + x^2/(x^2 - y^2).
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Factor (x^2 - y^2). We know that (x^2 - y^2) = (x + y)(x - y). You can check this by using the FOIL method, multiplying the first, outer, inner, and last terms. x^2 - xy + xy - y^2 = x^2 - y^2.
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Rewrite the problem as x/(x + y) + x^2/(x + y)(x - y).
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Find a common denominator: (x - y) and (x+ y)(x -y) have (x + y)(x - y) as a common denominator. You can see this by multiplying the first expression x/(x - y) by (x + y)/(x+ y) to get x(x + y)/(x + y)(x - y) + the second term x^2/(x + y)(x - y).
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Combine the terms under the new denominator to get [x(x + y) + x^2]/(x + y)(x - y).
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Factor out the parentheses in the numerator: [x(x + y) + x^2]/(x + y)(x - y) becomes (x^2 + xy + x^2)/(x + y)(x - y).
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Combine like terms: (2x^2 + xy)/(x + y)(x - y). This is the final answer.
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Take another example: 2/(x - y) - (3x + y)/(x^2 - y^2). This problem asks you to subtract algebraic fractions.
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Factor out x^2 - y^2 to get (x + y)(x - y). You can check this by using the FOIL method to factor out the parentheses. Multiplying the first, outer, inner and last terms of (x + y)(x - y), you get x^2 - xy + xy - y^2 = x^2 - y^2.
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Rewrite the problem as 2/(x - y) - (3x + y)/(x + y)(x - y).
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Find the common denominator: (x + y)(x - y) is the common denominator. You can see this by multiplying the first term 2/(x - y) by (x + y)/(x + y) to get 2(x + y)/(x - y)(x + y).
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Rewrite the problem as 2(x + y)/(x - y)(x + y)- (3x + y)/(x + y)(x - y).
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Combine the terms under the common denominator. [2(x + y) - (3x - y)]/(x - y)(x + y).
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Factor out the parentheses in the numerator: (2x + 2y - 3x - y)/(x - y)(x + y). Note that the last term y is preceded by a minus sign. This is due to the entire second term (3x + y) being preceded by a minus sign.
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Combine like terms (y - x)/(x - y)(x + y). This is the final answer.
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Tips & Warnings
When subtracting an algebraic expression, remember that the entire term takes the preceding minus sign. For example 2x + 3 - (4y + 2z) becomes 2x + 3 - 4y - 2z.
