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How to Algebraically Solve Fractions With Variables

Rational expressions and rational equations both contain fractions with variables in the denominators. Equations, unlike expressions, contain an equals sign that can be used to solve for the variable. Expressions can only be simplified or evaluated, and the latter only if a value for the variable is provided. Solving a rational equation works similarly to other equations in that algebra is used to move terms away from the variable until it is isolated on one side.

Instructions

- 1
  Solve the rational equation (5 / (x + 2)) + (2 / x) = (3 / 5x). Begin by finding the least common denominator. Since x appears in the other two denominators, disregard it and multiply the other two together to form the LCD: (x + 2) * 5x = 5x(x + 2).
- 2
  Convert the fractions to the LCD: (5 / (x + 2)) * (5x / 5x) = (25x / 5x(x + 2)); (2 / x) * ((5(x + 2) / 5(x + 2)) = ((10x + 20) / (5(x + 2)); and (3 / 5x) * ((x + 2) / (x + 2)) = ((3x + 6) / (5x(x + 2)).
- 3
  Disregard the denominators, since they are now all equal, and rewrite the numerators in terms of the original equation: (25x) + (10x + 20) = 3x + 6. Combine the like terms on the left side: 35x + 20 = 3x + 6. Subtract 20 from both sides: 35x = 3x + -14. Subtract 3x from both sides: 32x = - 14, and divide both sides by 32: x = -14 / 32 or x = - 7 / 16.

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