We will learn how to multiply a linear expression by a number, divide a linear expression by a number, and remove parentheses using the distributive property. These are fundamental operations with algebraic expressions and will be used many times from now on.
Multiplying and Dividing a Single-Term Expression by a Number
(Number) × (Coefficient × Letter): multiply the numbers, and leave the letter unchanged.
Example: 4 × 3x ＝ 12x, (−2) × 5a ＝ −10a
(Coefficient × Letter) ÷ Number: divide the coefficient by the number (or write it as a fraction).
Example: 8x ÷ 4 ＝ 2x, 12a ÷ (−3) ＝ −4a, 5x ÷ 2 ＝ 5x/2
Multiplying and Dividing a Linear Expression by a Number (Distributive Property)
When multiplying a linear expression by a number, multiply every term inside the parentheses (the distributive property).
a × (b + c) ＝ ab + ac
- 3 × (2x + 5) ＝ 3×2x + 3×5 ＝ 6x + 15
- (−4) × (3a − 7) ＝ (−4)×3a + (−4)×(−7) ＝ −12a + 28
When dividing a linear expression by a number, divide each term by the number as well (when using a fraction, multiply every term by 1/(number)).
Example: (8x − 6) ÷ 2 ＝ 8x/2 − 6/2 ＝ 4x − 3
Calculate the following.
(1) 5(2x − 3) (2) (−3)(4a + 6) (3) (15x − 10) ÷ 5
Answer:
(1) 5×2x + 5×(−3) ＝ 10x − 15
(2) (−3)×4a + (−3)×6 ＝ −12a − 18
(3) 15x÷5 − 10÷5 ＝ 3x − 2
Calculating Expressions Containing Parentheses
Remove the parentheses using the distributive property, then combine like terms. Follow these steps.
① Remove the parentheses (distributive property)
② Combine like terms
Calculate the following: 2(3x + 1) − 4(x − 5)
Answer:
① Distribute: 2×3x + 2×1 − 4×x − 4×(−5) ＝ 6x + 2 − 4x + 20
② Combine like terms: (6x − 4x) + (2 + 20) ＝ 2x + 22
- Linear expression × number: multiply every term inside the parentheses by the number (distributive property)
- Linear expression ÷ number: divide each term by the number (you may also multiply each term by 1/number as a fraction)
- When multiplying by a negative number, watch for changes in signs. The signs of all terms change
- Calculate in the order: remove parentheses → combine like terms
Practice Problems
- Calculate the following: (−4)(2x − 3)
- Calculate the following: (20a − 16) ÷ (−4)
- Calculate the following: 3(2x − 5) + 2(−x + 4)
Answers and Explanations
- Answer: −8x + 12
Explanation: (−4)×2x = −8x, (−4)×(−3) = +12. Therefore, −8x + 12. Watch for the change in signs when multiplying by a negative number. - Answer: −5a + 4
Explanation: Divide each term by −4. 20a ÷ (−4) = −5a, −16 ÷ (−4) = +4. Therefore, −5a + 4. - Answer: 4x − 7
Explanation: Distribute → 6x − 15 − 2x + 8. Like terms: (6x − 2x) + (−15 + 8) = 4x − 7.