### Key Concepts

• Solve equations using the distributive property

• Solve equations using distributing a negative number

• Solve equations using distributing a rational number

**5.3 Solve Equations Using the Distributive Property**

**Distributive property:**

The distributive property involves the use of parentheses and explains how to multiply a number or term outside the parentheses with the numbers or terms inside the parentheses.

#### Steps for Solving Algebra Equations using distributive property:

- If you see parenthesis with more than one term inside, then distribute first!
- Rewrite your equations with like terms together. Take the sign in front of each term.

- Combine like terms.
- Continue solving the one or two-step equations.

**Example:**

**5.3.1. Solve Equations Using the Distributive Property**

**Example 1:**

Solve equation using the distributive property.

2(5x – 3) = 14

**Sol:**

2(5x – 3) = 14

(2 × 5x) – (2 × 3) =14

10x – 6 =14

10x – 6 + 6 = 14 + 6 Add 6 to both sides

10x = 20 Divide 10 by each side

x = 2

**5.3.2. Solve Equations Using Distributing a Negative Number**

#### Negative Number:

A negative number represents the opposite. In the real number system, a negative number is a number that is less than zero.

**Example 1:**

Use the distributive property to solve the equation.

–6(m – 3) = –30

**Sol:**

–6(m – 3) = –30

–6m + 18 = –30

–6m + 18 – 18 = –30 – 18 Subtract 18 from both sides

–6m = –48 Divide each side by 6

m = 8

**5.3.3. Solve Equations Using Distributing a Rational number**

#### Rational number:

A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator *p* and a non-zero denominator *q*.

**Example 1:**

1818

(p + 24) =9

**Sol:**

1818

(p + 24) =9

1818

p +

1818 (24) =9

18

P + 3 =9

18

P + 3 – 3 =9 – 3 Subtract 3 from both sides.

18

P = 6

( 8 1)1 8( 8 1)1 8

P = 6 (

8 1)8 1)

P = 48

## Exercise:

1. Solve the equation using distributive property – 4(x + 3) = 8.

2. A gym charges a $50 activation fee and $17 per month for a membership. If you spend $356, for how many months do you have a gym membership?

3. Solve the equation 3x + 2(2x -1) = 33.

4. Use the distributive property to solve the equation 6(x + 3) = 48.

5. Solve the equation -2 (x – 2) = 4. 3

6. Solve the equation using distributive property -106 = -2(5 + 6x)

7. 3(x + 3) = -15

8. 0.4(x – 0.45) = 9.2

9. A family of 7 bought tickets to the circus. Each family member also bought a souvenir that cost $6. The total amount they spent was $147. How much did one ticket cost?

10. Use the distributive property to solve the equation -2(p – 200) =42.

### Concept Map

### What have we learned:

■Understand the distributive property

■ Understand how to solve equations using the distributive property

■ Solve equations using distributing a negative number

■ Solve equations using distributing a rational number

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