0580

Linear Equations and Inequalities

Algebra and Sequences

Brackets

  • Expand the bracket first, then solve as usual
    • 3(2x + 5) = 39 expands to 6x + 15 = 39, giving 6x = 24 and x = 4
  • Dividing through by the number outside also works, though it can create fractions

Fractions

  • Scale the whole equation by the lowest common denominator, so no term is left divided
  • Terms without a denominator must be multiplied too, which is the step most often missed
Worked example

Clearing fractions

Solve x/3 + x/4 = 7.

Solution:

  • The lowest common denominator of 3 and 4 is 12
  • Scaling every term by 12 gives 4x + 3x = 84
  • Collect: 7x = 84
  • x = 12

Unknowns on both sides

  • Remove the smaller unknown term by applying its inverse to both sides, which avoids negatives
    • 7x − 4 = 2x + 21 becomes 5x − 4 = 21 after subtracting 2x
    • Then 5x = 25 and x = 5
  • Fractional equations with algebraic denominators are handled the same way, and are covered under algebraic fractions