Making Algebra Easier with the Distributive Property

When you're trying to simplify algebraic expressions, one really helpful tool is the Distributive Property.

This tool lets you take a number outside a set of parentheses and multiply it by each number inside. This way, tough expressions become much simpler! Let’s go through the steps with some examples to make it clear.

What is the Distributive Property?

The Distributive Property says that for any numbers a, b, and c, this equation works:

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

This means you can take a and multiply it by both b and c. It’s a simple idea that’s super useful in algebra!

How to Use the Distributive Property

Here’s how to apply it in three easy steps:

1. Find the Terms: Look for the parts inside the parentheses that you need to work with.
2. Multiply Each Term: Take the number outside the parentheses and multiply it by each part inside.
3. Combine Like Terms: After multiplying, put together any similar terms to make the expression simpler.

Example 1: Distributing a Single Term

Let’s try this expression:

$$3(x + 4)$$

In this example, 3 is outside the parentheses. We’ll use the Distributive Property to multiply 3 by each part inside:

$$3(x) + 3(4) = 3x + 12$$

Example 2: Distributing with Multiple Terms

Now, let’s look at a trickier example:

$$2(3x + 5y) - 4(2x - 3y)$$

Here, we’ll distribute 2 across 3x + 5y and -4 across 2x - 3y.

• First, let’s distribute 2:

  $$2(3x) + 2(5y) = 6x + 10y$$

• Now, let’s distribute -4:

  $$-4(2x) + (-4)(-3y) = -8x + 12y$$

Now, let’s put everything together:

$$6x + 10y - 8x + 12y$$

Combine Like Terms

Next, we need to combine the similar terms:

$$(6x - 8x) + (10y + 12y) = -2x + 22y$$

Conclusion

The Distributive Property makes it much easier to simplify expressions. It’s also important for solving equations later on.

Practice using it with different expressions, and you’ll feel more confident in your algebra skills! Just remember to distribute carefully and combine those like terms step by step. Happy simplifying!