Using the Distributive Property to Write Equivalent Expressions

7.8.2 Using the Distributive Property to Write Equivalent Expressions

7.8.2 • Using the Distributive Property to Write Equivalent Expressions

Activity

Applying the distributive property to multiply out the factors of, or expand, $4 (x + 2)$ gives us $4 x + 8$ , so we know the two expressions are equivalent. We can use a rectangle with side lengths $(x + 2)$ and 4 to illustrate the multiplication.

An area model using a rectangle is divided into 2 smaller rectangles with a vertical segment. The top left section of the rectangle is labeled x and the top right of the rectangle is labeled 2. The side length of the rectangle is labeled 4. There are two areas inside the rectangle: The left portion of the rectangle bordered by the x and 4 is labeled 4 times x. The right portion of the rectangle bordered by the 2 (on top) is labeled with an 8.

1. Draw a diagram to show that $n (2 n + 5)$ and $2 n^{2} + 5 n$ are equivalent expressions.

Compare your answer:

An area model using a rectangle is divided into 2 smaller rectangles with a vertical segment. The top left section of the rectangle is labeled 2 times n and the top right of the rectangle is labeled 5. The side length of the rectangle is labeled n. There are two areas inside the rectangle: The left portion of the rectangle bordered by the 2n and n is labeled 2 times n squared. The right portion of the rectangle bordered by the 5 (on top) is labeled with an 5 times n.

2. For each expression, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.

a. $6 (\frac{1}{3} n + 2)$

Compare your answer:

$2 n + 12$

b. $p (4 p + 9)$

Compare your answer:

$4 p^{2} + 9 p$

c. $5 r (r + \frac{3}{5})$

Compare your answer:

$5 r^{2} + 3 r$

d. $(0.5 w + 7) w$

Compare your answer:

$0.5 w^{2} + 7 w$

Self Check

Which expression is equivalent to

4t(t-\frac{3}{4})

$4t^2-\frac{3}{4}$
$4t^2-3t$
$4t^2-\frac{3}{4}t$
$4t^2-3$

Additional Resources

Distributive Property

Distributive Property

If $a$ , $b$ , $c$ are real numbers, then:

$a (b + c) = a b + a c$

Other forms:

$a (b - c) = a b - a c$

$(b + c) a = b a + c a$

Example 1

Simplify $3 (x + 4)$

Step 1 - Distribute the 3 to both the $x$ and the 4.

Diagram showing 3 multiplied times the quantity of x plus 4. An arrow is going from 3 to x. Another arrow is going from 3 to 4.

Step 2 - Multiply the terms.

$3 (x) + 3 (4)$

Step 3 - Simplify.

$3 x + 12$

Example 2

Simplify $m (7 - 2 m)$

Step 1 - Distribute the $m$ to both the 7 and the $2 m$ .

Diagram showing m multiplied times the quantity of 7 minus 2 times m. An arrow is going from m to 7. Another arrow is going from m to 2 times m.

Step 2 - Multiply the terms. Recall $m \cdot m = m^{2}$ .

$m (7) - m (2 m)$

Step 3 - Simplify.

$7 m - 2 m^{2}$

Example 3

Simplify $5 x (x - \frac{2}{5})$

Step 1 - Distribute the $5 x$ to both the $x$ and the $\frac{2}{5}$ .

$Diagram showing 5 times x multiplied times the quantity of x minus the fraction two-fifths. An arrow is going from 5 times x to x. Another arrow is going from 5 times x to two-fifths.$

Step 2 - Multiply the terms. Recall $x \cdot x = x^{2}$ .

$5 x (x) - 5 x (\frac{2}{5})$

Step 3 - Simplify.

$5 x^{2} - 2 x$

Try it

Distributive Property

Simplify $7 y (4 y - \frac{3}{7})$

Here is how to simplify the expression using the distributive property:

Step 1 - Distribute the $7 y$ to both the $4 y$ and the $\frac{3}{7}$ .

$Diagram showing 7 times y multiplied times the quantity of 4 times y minus the fraction three-sevenths. An arrow is going from 7 times y to 4 times y. Another arrow is going from 7 times y to three-sevenths.$

Step 2 - Multiply the terms. Recall $y \cdot y = y^{2}$ . $7 y (4 y) - 7 y (\frac{3}{7})$

Step 3 - Simplify. $28 y^{2} - 3 y$