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1 Algebra
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Australia
Year 10

1.01 Adding and subtracting algebraic fractions

Lesson
Worksheet
Practice
Lesson

Addition and subtraction of algebraic fractions

The most important things to remember when adding and subtracting fractions:

  • We need like denominators.

  • We need to keep our fractions equivalent.

Now we are going to build on this knowledge and look at how to add and subtract algebraic fractions.

Examples

Example 1

Fully simplify the following expression: \dfrac{6x}{2}-\dfrac{7x}{2}

Worked Solution
Create a strategy

We can write the numerators together as a single expression over the common denominator.

Apply the idea
\displaystyle \dfrac{6x}{2}-\dfrac{7x}{2}\displaystyle =\displaystyle \frac{6x-7x}{2}Subtract the numerators
\displaystyle =\displaystyle -\dfrac{x}{2}Evaluate the subtraction

Example 2

Simplify the following: \dfrac{3x}{5}-\dfrac{x}{7}

Worked Solution
Create a strategy

Find a common denominator to add the fractions.

Apply the idea

5\times 7=35 so 35 is a common multiple of the denominators. So we need to multiply the numerator and denominator of each fraction by the other denominator.

\displaystyle \dfrac{3x}{5}-\dfrac{x}{7}\displaystyle =\displaystyle \frac{3x \times 7}{5 \times 7} - \frac{x \times 5}{7 \times 5}Multiply fractions by the other denominator
\displaystyle =\displaystyle \frac{21x}{35}-\frac{5x}{35}Evaluate the second fraction
\displaystyle =\displaystyle \frac{21x-5x}{35}Subtract the numerators
\displaystyle =\displaystyle \frac{16x}{35}Evaluate the subtraction

Example 3

Simplify the following: \dfrac{-3x-1}{6}+\dfrac{x+6}{6}

Worked Solution
Create a strategy

We can write the numerators together as a single expression over the common denominator.

Apply the idea
\displaystyle \dfrac{-3x-1}{6}+\dfrac{x+6}{6}\displaystyle =\displaystyle \dfrac{(-3x-1) + (x+6)}{6}Add the numerators
\displaystyle =\displaystyle \dfrac{-3x+x-1+6}{6}Collect like terms
\displaystyle =\displaystyle \dfrac{-2x+5}{6}Subtract the numerator
\displaystyle =\displaystyle -\dfrac{2x+5}{6}Simplify
Idea summary

The most important things to remember when adding and subtracting fractions:

  • We need like denominators.

  • We need to keep our fractions equivalent.

Outcomes

ACMNA232

Apply the four operations to simple algebraic fractions with numerical denominators

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