Problem 3 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$-4 y+5 y$$

Step-by-Step Solution

Answer

The combined expression is simply y.

Step 1: Identify Like Terms

Identify the terms in the expression that can be combined because they have the same variable part. Here, both -4y and 5y have 'y' as their variable component.

Step 2: Apply the Distributive Property

Use the distributive property by factoring out the common variable from the two terms. Since both terms include 'y', rewrite the expression as: (-4 + 5)y.

Step 3: Perform Arithmetic

Calculate the sum inside the parenthesis: -4 + 5 = 1.

Step 4: Write the Final Expression

With the result of the arithmetic operation, write the final expression by distributing 'y' back: 1y or simply y.

Key Concepts

Understanding Like Terms

In algebra, 'like terms' are pivotal because they allow us to simplify expressions with ease. Like terms are terms that have identical variables raised to the same power. They can differ in their coefficients, the numerical part of the term. Understanding this allows us to combine them by performing arithmetic operations on these coefficients.
For example, in the expression

• -4y
• 5y

the variable part 'y' is the same, making them like terms. This concept is crucial because only like terms can be combined using addition or subtraction to simplify algebraic expressions. This forms the basis for more complex operations in algebra, so grasping it early is essential.

Combining Similar Terms: Simplifying Expressions

Combining similar terms is an essential technique in algebra to simplify expressions. When you identify like terms, you can combine them to make your expressions neater and reduce complexity. This process is akin to collecting similar items, making problem-solving more manageable.
Take the expression

• -4y + 5y

Since both terms have the variable 'y', we can add their coefficients: -4 and 5. It's like having -4 apples and gaining 5 more; in total, you would have 1 apple. Similarly, after combining these similar terms, the expression becomes y.
This simplification not only makes expressions easier to understand but provides a foundation for more advanced algebraic operations.

Factoring in Algebra: The Distributive Property

Factoring in algebra often involves using the distributive property to simplify expressions. This property states that you can multiply a sum by a single number to achieve the same effect as multiplying each addend individually and then summing them.
In the example with

• -4y + 5y

we applied the distributive property by extracting the common variable 'y', reimagining it as i.e., (-4 + 5)y.
Here, the factoring step simplifies the process of combining the terms by first dealing with their numerical parts. After calculating

1. -4 + 5 = 1

we multiply the result by 'y'. Recognizing this property reinforces algebraic manipulation and makes handling more complex equations and expressions significantly easier.