In this lesson, we will be using the distributive property to expand algebraic expressions. Algebra tiles are used to show the concepts involved.
Two more examples of how to expand a problem using the distributive property.
This video uses the distributive property to create equivalent fractions and then gives two examples of how this can be helpful when simplifying expressions.
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Review:
Distributive Property of Multiplication Over Addition
states that a product can be found by adding, then multiplying, or by multiplying, then adding.
Example 1
| $3 * (10 + 4)$ | $=$ | $(3 * 10) + (3 * 4)$ |
| $3 * (13)$ | $=$ | $(30) + (12)$ |
| $42$ | $=$ | $42$ |
Example 2
| $3 * (n + x)$ | $=$ | $(3 * n) + (3 * x)$ |
| $3(n + x)$ | $=$ | $(3n) + (3x)$ |
Try these four problems:
Fill in the blanks:
1. $7(5+3)=$______ $+$______
2. $6a+6b=$______$(a+b)$
Simplify the expressions:
3. $2x + 3(x-4 + 2x) + (2-2x)$
4. $\frac{x + 1}{5n} + \frac{4-2x}{5n}$
Answers:
| 1. Original problem: | $7(5+3)=$____ $+$____ |
| a. Distribute the 7 | $7(5+3)=$ $7*5$$ + $$7*3$ |
| ………………………………….. | ……………………………. |
| 2. Original problem: | $6a+6b=$______$(a+b)$ |
| a. Factor out the 6 | $6a+6b=$ $6$$(a+b)$ |
| ………………………………….. | ……………………………. |
| 3. Original problem: | $2x + 3($$x$$- 4 +$$2x$$) + (2-2x)$ |
| a. Simplify what we can in the parentheses | $2x+\color{#f15a23}{3(}\color{blue}{3x}-4\color{#f15a23})\color{#4a4949″}{+(2-2x)}$ |
| b. Use the distributive property to remove the parentheses | $2x + $$9x-12$$-4 + 2-2x$ |
| c. Combine like terms | $\color{blue}{2x + 9x} $$- 12 + 2$$ -2x $ |
| d. Solution | $\color{red}{9x-10}$ |
| ………………………………….. | ……………………………. |
| 4. Original problem: | $\dfrac{x + 1}{5n} + \dfrac{4-2x}{5n}$ |
| a. Since the fractions have a common denominator we can add them. | $\dfrac{(x + 1) + (4-2x)}{5n}$ |
| b. Rearrange the terms. | $\dfrac{(x + ^-2x) + (1+4)}{5n}$ |
| c. Now we can combine like terms | $\color{red}{\dfrac{5-x}{5n}}$ |
