5
Adding and subtracting algebraic fractions
Textbook: pp. 22–26
GoalAdd and subtract algebraic fractions with equal and different denominators; reduce the result.
New words
adding fractions · kasrlarni qo‘shishsubtracting fractions · kasrlarni ayirishcommon denominator · umumiy maxrajreducing the result · natijani qisqartirish
Explanation
Fractions with equal denominators are added by adding the numerators and keeping the denominator: a/c + b/c = (a + b)/c. Subtraction works the same way: a/c - b/c = (a - b)/c, but the numerator of the subtrahend must be put in brackets, otherwise the signs go wrong. If the denominators differ, first bring the fractions to a common denominator, then add or subtract the numerators. At the end factor the numerator and denominator and try to reduce the fraction. Denominators are never added: 1/a + 1/b ≠ 2/(a + b).
Worked examples
a/(a - b) - b/(a - b) = (a - b)/(a - b) = 1, where a ≠ b.
2/(x + 1) + 3/(x - 1) = (2(x - 1) + 3(x + 1))/((x + 1)(x - 1)) = (2x - 2 + 3x + 3)/(x² - 1) = (5x + 1)/(x² - 1).
Class activity
“Fraction relay”: the first pupil in a team brings two fractions to a common denominator, the second adds, the third reduces and the fourth checks by substituting a number.
Practice
1
Compute 3/(2a) + 5/(2a).
4/a
2
Write 1/x - 1/y as one fraction.
(y - x)/(xy)
3
Find the value of 2/(x + 1) + 3/(x - 1) for x = 3.
2
4
For a = b = 1, what are the values of 1/a + 1/b and 2/(a + b)? What follows?
The first is 2, the second is 1; so 1/a + 1/b ≠ 2/(a + b) and denominators cannot be added.