How to Solve Adding Fractions With Variables Attached

Adding Algebraic Expressions with a Common Denominator

• 1

  Add the numerators of all expressions. For example, if given, (3x)/(2x + 1) + (5x -3)/(2x + 1) + (2x + 8)/ (2x + 1), write 3x + 5x - 3 + 2x + 8. Perform the addition. This yields 3x + 5x + 2x - 3 + 8, or 10x + 5.

• 2

  Write the new expression with the sum of the numerators over the common denominator. In the example: (10x + 5) / (2x + 1).

• 3

  Simplify the result. Factoring and reducing to lowest terms gives:

  (10x + 5) / (2x + 1) = [5(2x + 1)] / (2x + 1) = 5.

Adding Algebraic Expressions without a Common Denominator

• 4

  Find the least common denominator (LCD) of the terms by taking the least common multiple (LCM) of the individual denominators and multiplying them. For example, if given: [3 / (x + 2)] - [(2x) / (x - 3)], the LCD is (x + 2)(x - 3).

• 5

  Rewrite each expression with the LCD. To achieve this and not change the values of any of the expressions, each expression must be multiplied by the LCD in the numerator and denominator.

  In the example:

  3 / (x + 2) becomes 3 (x - 3) / [(x + 2) (x - 3)] and 2x / (x - 3) becomes [2x (x + 2)] / [(x + 2) (x - 3)].

• 6

  Add the numerators, performing all necessary multiplication and addition, combining like terms and writing the numerator in standard form. Keep the denominator the same.

  In the example:

  [3 (x - 3)] / [(x + 2) (x - 3)] + [2x (x + 2)] / [(x + 2) (x - 3)] =

  [3x - 9 + 2x^2 + 4x] /[ (x - 3) ( x + 2)] =

  [2x^2 + 7x - 9] /[ (x - 3) ( x + 2)].