Example Of A Like Term

Understanding Like Terms: A Comprehensive Guide with Examples

Understanding like terms is fundamental to mastering algebra and other higher-level mathematical concepts. This comprehensive guide will delve into the definition of like terms, explore various examples, explain the significance of identifying them, and address frequently asked questions. By the end, you'll be confident in recognizing and manipulating like terms in your mathematical endeavors.

What are Like Terms?

Like terms, in algebra, are terms that have the same variables raised to the same powers. The coefficients (the numbers in front of the variables) can be different, but the variables and their exponents must be identical for terms to be considered like terms. Think of it like sorting LEGO bricks – you can only combine bricks that are exactly the same shape and size. Similarly, you can only combine like terms in algebraic expressions.

Examples of Like Terms

Let's look at some examples to solidify our understanding. Consider the following expressions:

• Example 1: 3x + 5x

Here, both terms, 3x and 5x, contain the variable 'x' raised to the power of 1 (remember, x is the same as x¹). Therefore, 3x and 5x are like terms.

• Example 2: 7y² - 2y² + y²

In this example, all three terms contain the variable 'y' raised to the power of 2 (y²). Therefore, 7y², -2y², and y² are all like terms.

• Example 3: 4ab + 6ab - 2ab

These terms all contain the variables 'a' and 'b', each raised to the power of 1. The order of the variables doesn't matter; as long as the variables and their exponents are the same, the terms are like terms.

• Example 4: 2x³y²z + 5x³y²z - x³y²z

All terms contain x³, y², and z. Therefore, these are like terms.

• Example 5: 1/2 xy and 2/3 xy

Even fractional coefficients don't prevent terms from being like terms. Both terms contain 'x' and 'y' each raised to the power of 1.

Examples of Unlike Terms

Now, let's look at examples of unlike terms to highlight the differences:

• Example 1: 2x and 2y

These terms contain different variables ('x' and 'y'), making them unlike terms.

• Example 2: 3x² and 3x

Even though they both have 'x', the exponents are different (2 and 1). This makes them unlike terms.

• Example 3: 5ab and 5a²b

Here, the variable 'a' has different exponents (1 and 2), resulting in unlike terms.

• Example 4: 4xy and 4xyz

The inclusion of an extra variable ('z') in the second term makes them unlike terms.

• Example 5: 6 and 6x

One term is a constant (a number without a variable), while the other has a variable. Constants are only like terms with other constants.

Why is Identifying Like Terms Important?

Identifying like terms is crucial because it allows us to simplify algebraic expressions. We can only combine like terms through addition or subtraction. This simplification process makes algebraic equations easier to solve and understand. Consider this example:

3x + 5x + 2y - y

We can combine the like terms (3x and 5x) and (2y and -y):

(3x + 5x) + (2y - y) = 8x + y

Without identifying like terms, we wouldn't be able to simplify this expression.

Combining Like Terms: A Step-by-Step Guide

Combining like terms involves adding or subtracting the coefficients of the like terms while keeping the variables and their exponents the same.

Step 1: Identify the like terms. Carefully examine the expression and group the terms that have the same variables raised to the same powers.

Step 2: Add or subtract the coefficients of the like terms. Perform the addition or subtraction operation on the coefficients.

Step 3: Write the simplified expression. Combine the results from step 2 with the common variables and their exponents.

Example: Simplify the expression: 7a²b + 3ab² - 2a²b + 5ab²

Step 1: Identify like terms: 7a²b and -2a²b are like terms. 3ab² and 5ab² are like terms.

Step 2: Combine like terms: (7a²b - 2a²b) = 5a²b (3ab² + 5ab²) = 8ab²

Step 3: Write the simplified expression: 5a²b + 8ab²

Notice that we cannot combine 5a²b and 8ab² because they are unlike terms (different exponents).

Like Terms in Different Contexts

The concept of like terms extends beyond simple algebraic expressions. You'll encounter them in various mathematical contexts:

• Polynomials: Polynomials are expressions consisting of variables and constants. Identifying like terms is essential for simplifying and manipulating polynomials, such as adding, subtracting, and multiplying them.
• Equations: Solving equations often involves simplifying expressions by combining like terms. This simplification helps isolate the variable and find its value.
• Inequalities: Similar to equations, simplifying inequalities often necessitates identifying and combining like terms.

Advanced Examples and Applications

Let's explore more complex examples demonstrating the application of like terms:

Example 1: Simplify (3x²y + 2xy² - 5x²y + 7xy²)

Like terms: 3x²y and -5x²y; 2xy² and 7xy²

Simplified expression: (3x²y - 5x²y) + (2xy² + 7xy²) = -2x²y + 9xy²

Example 2: Solve the equation: 4x + 6 = 2x + 10

First, we need to rearrange the equation to get like terms on the same side:

2x = 4 (by subtracting 2x from both sides and subtracting 6 from both sides)

x = 2 (by dividing both sides by 2)

Example 3: Simplify (1/3)a²b³ + (2/5)a²b³ - (1/2)a²b³

Find a common denominator (30) to add the fractions:

(10/30)a²b³ + (12/30)a²b³ - (15/30)a²b³ = (7/30)a²b³

Frequently Asked Questions (FAQ)

Q1: Are constants like terms?

A1: Yes, all constants (numbers without variables) are like terms. For instance, 5, 10, and -2 are all like terms.

Q2: Can I combine like terms with different coefficients?

A2: Absolutely! The coefficients can be different; you only need the variables and their exponents to be identical. You add or subtract the coefficients accordingly.

Q3: Does the order of variables matter when identifying like terms?

A3: No, the order of variables does not matter. For example, 2xy and 2yx are like terms.

Q4: What happens if there are no like terms in an expression?

A4: If there are no like terms, the expression is already in its simplest form, and no further simplification is possible.

Q5: How are like terms used in solving word problems?

A5: Word problems often involve translating real-world scenarios into algebraic expressions. Identifying and combining like terms is crucial for simplifying these expressions and solving for unknown variables.

Conclusion

Mastering the concept of like terms is a cornerstone of algebraic proficiency. By understanding their definition, recognizing them in different contexts, and applying the techniques for combining them, you'll build a strong foundation for more advanced mathematical concepts. Remember, the key is to carefully examine the variables and their exponents – they must be identical for terms to be considered like terms. Consistent practice and attention to detail will make you confident in your ability to simplify and manipulate algebraic expressions. Through continuous learning and application, you'll not only master like terms but also unlock the broader world of algebra and its many applications.