Introduction

In mathematics, the distributive property is a basic and essential rule utilized extensively in solving equations. It refers to a principle stating that when you have a value multiplied by an expression in parentheses, such as, a (b + c), this equates to the sum of two separate products, namely a * b + a * c. This fundamental property proves relevant and useful in various mathematical processes, including solving equations and simplifying expressions.

In this article, we will discuss four applicable strategies that leverage the distributive property when solving different types of equations.

1. Simplifying Expressions

When dealing with complicated expressions involving parentheses, the distributive property can simplify these expressions making it easier to solve the equation. Here’s an example:

5(x + 6) = 3(x + 4) + 2

To simplify this expression, use the distributive property:

5x + 30 = 3x + 12 + 2

From here, you can solve the equation more comfortably by combining like terms on both sides of the equation:

5x – 3x = 14 -30

2x = -16

x = -8

2. Combining Like Terms within Parentheses

If you come across an equation containing expressions with similar variables within parentheses, apply the distributive property as follows.

Example:

(3x + 8) + (7x + 4) = (11y -6)

Combine like terms inside both sets of parentheses by adding them together:

10x+12 = 11y -6

Using this simplified form makes solving subsequent equations much easier.

3. Factoring Out Common Factors

Another helpful strategy is identifying common factors on both sides of an equation and subsequently factoring them out.

Example:

18x – 27y = 24z – 36

In this case, you would recognize that the common factor is 3 for both sides:

3 (6x – 9y) = 3 (8z -12)

Divide both sides by 3:

6x – 9y = 8z -12

Now you have a simplified equation to work with when trying to solve for x, y, or z.

4. Solving Quadratic Equations

The distributive property is also beneficial for handling quadratic equations.

Example:

(x+4)(x-2)=0

By applying the distributive property to the left part of the equation, you get:

x^2 + 4x – 2x -8 = 0

Combine like terms:

x^2 + 2x -8 =0

Now you’re set up for familiar quadratic solving steps like factoring, completing the square, or utilizing the quadratic formula.

Conclusion

The distributive property serves as a valuable resource to make equations less complicated and simplify expressions. Utilizing these four strategies involving the distributive property will allow you to solve a wide range of equations more efficiently. Keep practicing these techniques, and soon applying the distributive property will become second nature in your mathematical problem-solving journeys.