Have you ever encountered algebraic expressions that look complicated and difficult to solve? One of the fundamental concepts in algebra that makes simplification of expressions easier is “like terms”. Like terms are terms that have the same variable(s) raised to the same power(s). For example, 2x and 3x are like terms, but 2x and 3x^2 are not like terms. Understanding like terms is crucial in algebra as it simplifies expressions and makes them easier to work with. In this article, we’ll explore the characteristics of like terms, how to add and subtract expressions with like terms, and common mistakes to avoid while working with them.
Characteristics of like terms
Like terms have two important characteristics: the same variables and the same powers. Let’s take a look at a few examples:
- 3x and 5x are like terms because they have the same variable ‘x’ raised to the power 1.
- 4y^2 and 2y^2 are like terms because they have the same variable ‘y’ raised to the power 2.
- 7z and 10z are like terms because they have the same variable ‘z’ raised to the power 1.
It’s important to note that the coefficients (numbers in front of the variables) can be different, but the variables and their powers must be the same for two terms to be considered like terms.
Addition of like terms
Adding like terms is simple. You just add the coefficients (numbers in front of the variables) and keep the variables and their powers the same. Let’s take a look at some examples:
- 2x + 3x = (2+3)x = 5x
- 4y^2 + 2y^2 = (4+2)y^2 = 6y^2
- 7z + 10z = (7+10)z = 17z
You can see that in each case, we simply added the coefficients and kept the variables and their powers the same.
Examples of addition of like terms
Let’s look at some more examples to solidify our understanding of adding like terms.
Example 1: 5x + 3x – 2x
We have three terms here: 5x, 3x, and -2x. The first two terms are like terms because they have the same variable ‘x’ raised to the power 1. We can add them first:
5x + 3x = (5+3)x = 8x
Our expression now becomes:
8x – 2x
The last term is also a like term with the same variable ‘x’ raised to the power 1, so we can add it to the previous result:
8x – 2x = (8-2)x = 6x
Therefore, 5x + 3x – 2x simplifies to 6x.
Example 2: 4y^2 + 3y + 2y^2 – y
Here, we have four terms: 4y^2, 3y, 2y^2, and -y. The first and the third terms are like terms because they have the same variable ‘y’ raised to the power 2. We can add them first:
4y^2 + 2y^2 = (4+2)y^2 = 6y^2
Our expression now becomes:
6y^2 + 3y – y
The second and the last terms are like terms because they have the same variable ‘y’ raised to the power 1. We can add them next:
3y – y = 2y
Therefore, 4y^2 + 3y + 2y^2 – y simplifies to 6y^2 + 2y.
Subtraction of like terms
Subtracting like terms is similar to adding them. Instead of adding the coefficients, we subtract them while keeping the variables and their powers the same. Let’s look at an example:
- 8x – 3x = (8-3)x = 5x
As you can see, we simply subtracted the coefficients and kept the variables and their powers the same.
Examples of subtraction of like terms
Let’s look at some examples of subtracting like terms to further understand the concept.
Example 1: 9a^2 – 5a^2
We have two terms here: 9a^2 and 5a^2. They are like terms because they have the same variable ‘a’ raised to the power 2. We can subtract them to get:
9a^2 – 5a^2 = (9-5)a^2 = 4a^2
Therefore, 9a^2 – 5a^2 simplifies to 4a^2.
Example 2: 7b^3 – 4b^3 – 2b^3
Here, we have three terms: 7b^3, -4b^3, and -2b^3. The first term is not like terms with the second and the third terms because they do not have the same coefficient. However, the second and the third terms are like terms because they have the same variable ‘b’ raised to the power 3. We can subtract them first:
-4b^3 – 2b^3 = (-4-2)b^3 = -6b^3
Our expression now becomes:
7b^3 – 6b^3
The first and the second terms are like terms because they have the same variable ‘b’ raised to the power 3. We can subtract them next:
7b^3 – 6b^3 = (7-6)b^3 = b^3
Therefore, 7b^3 – 4b^3 – 2b^3 simplifies to b^3.
Importance of understanding like terms in algebra
Understanding like terms is crucial in algebra because it simplifies expressions and makes them easier to work with. If we have an expression with unlike terms, we cannot add or subtract them directly. We need to simplify the expression first by grouping the like terms together. Once we have simplified the expression, we can add or subtract the like terms.
For example, let’s say we have the expression 3x + 2y – 5x – 3y. We cannot add or subtract these terms directly because they are not like terms. However, we can simplify the expression by grouping the like terms together:
3x – 5x + 2y – 3y = -2x – y
Now, we have an expression with like terms, and we can simplify it further if needed.
Common mistakes to avoid when working with like terms
One common mistake students make when working with like terms is to only consider the variables and forget about the coefficients. Remember, like terms must have the same variables and the same powers, but the coefficients can be different.
Another mistake students make is to only consider the variables and their powers and forget about the signs. Remember, the signs of the terms are important when adding or subtracting them.
A third mistake students make is to add or subtract unlike terms. Remember, unlike terms cannot be added or subtracted directly. We need to simplify the expression first by grouping the like terms together.
Practice problems for addition and subtraction of like terms
Let’s practice adding and subtracting like terms with some problems.
Problem 1: Simplify the expression 6x + 7x – 5x.
Solution:
6x + 7x – 5x = (6+7-5)x = 8x
Therefore, 6x + 7x – 5x simplifies to 8x.
Problem 2: Simplify the expression 8a^2 – 3a^2 + 5a^2.
Solution:
8a^2 – 3a^2 + 5a^2 = (8-3+5)a^2 = 10a^2
Therefore, 8a^2 – 3a^2 + 5a^2 simplifies to 10a^2.
Problem 3: Simplify the expression 4x^3 – 2x^2 + 3x^3 – 5x^3.
Solution:
4x^3 – 2x^2 + 3x^3 – 5x^3 = (4+3-5)x^3 – 2x^2 = 2x^3 – 2x^2
Therefore, 4x^3 – 2x^2 + 3x^3 – 5x^3 simplifies to 2x^3 – 2x^2.
Conclusion
Like terms are terms that have the same variables and the same powers. Adding and subtracting like terms is simple as we just add or subtract the coefficients and keep the variables and their powers the same. Understanding like terms is crucial in algebra as it simplifies expressions and makes them easier to work with. Remember to avoid common mistakes such as only considering the variables, forgetting the signs, or adding unlike terms. With practice, you’ll become more comfortable working with like terms and simplifying algebraic expressions.
