Lesson 2.8

Solving with Distribution

Parentheses are like a locked gate. Before you can solve for the variable inside, you must use the key of multiplication to unlock it.

Introduction

The Distributive Property allows us to remove parentheses by multiplying the term outside by every term inside. In , the 2 must be multiplied by the AND the 3.

Past Knowledge

You know that .

Today's Goal

Use distribution to simplify equations, then solve for the variable.

Future Success

Essential for solving complex inequalities and polynomial operations.

Key Concepts

The "Rainbow" Arrow Method

Draw arrows from the outside number to each term inside.

Multiply Multiply

Distributing Negatives

Be careful with signs! If you distribute a negative number, it flips the signs of everything inside.

  • (Negative times Negative is Positive!)

Worked Examples

Example 1: Basic Distribution

Basic

Solve for :

1

Distribute

Multiply 2 by and 2 by 5.

2

Solve Two-Step Equation

==
Divide by 2:

Example 2: Negative Distribution

Intermediate

Solve for :

1

Distribute Negative 3

Remember: .

2

Solve Two-Step Equation

==
Divide by -3:

Example 3: Distribute THEN Combine

Advanced

Solve for :

1

Distribute First

Multiply the 4 into the parentheses.

2

Combine Like Terms

Combine and .

3

Solve

==
Divide by 8:

Common Pitfalls

Forgetting the Second Number

The most common mistake! In , students write . You must distribute the 2 to the 3 as well ().

Distributing Over the Equals Sign

In , students sometimes try to distribute the 2 to the 10. No! The parentheses stop at the equals sign.

Real-Life Applications

Calculating total costs often involves groups. If buying tickets for 3 friends, and each ticket costs $20 plus an unknown service fee :

Distributing helps break down the total into "Total Ticket Price" + "Total Fees": .

Practice Quiz

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