Combine Like Terms: − 7 W 2 + 4 X 2 + 2 W X − 5 W 2 − 8 X 2 -7w^2 + 4x^2 + 2wx - 5w^2 - 8x^2 − 7 W 2 + 4 X 2 + 2 W X − 5 W 2 − 8 X 2 □ \square □

What are Like Terms?

In algebra, like terms are expressions that have the same variable(s) raised to the same power. They can have different coefficients, but the variable(s) and the exponent(s) must be the same. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Why Combine Like Terms?

Combining like terms is an essential skill in algebra that helps simplify complex expressions. It makes it easier to solve equations, graph functions, and perform other mathematical operations. By combining like terms, you can reduce the number of terms in an expression, making it more manageable and easier to work with.

Step-by-Step Guide to Combining Like Terms

To combine like terms, follow these steps:

Step 1: Identify the Like Terms

The first step is to identify the like terms in the expression. Look for terms that have the same variable(s) raised to the same power. In the given expression, $-7w^2 + 4x^2 + 2wx - 5w^2 - 8x^2$, the like terms are:

• $-7w^2$ and $-5w^2$ (both have the variable $w$ raised to the power of 2)
• $4x^2$ and $-8x^2$ (both have the variable $x$ raised to the power of 2)

Step 2: Combine the Coefficients

Once you have identified the like terms, combine their coefficients. To do this, add or subtract the coefficients of the like terms. In the given expression, the coefficients of the like terms are:

• $-7w^2$ and $-5w^2$: $-7 + (-5) = -12$
• $4x^2$ and $-8x^2$: $4 + (-8) = -4$

Step 3: Write the Combined Terms

After combining the coefficients, write the combined terms. In the given expression, the combined terms are:

• $-12w^2$
• $-4x^2$

Step 4: Simplify the Expression

Finally, simplify the expression by combining the combined terms. In the given expression, the simplified expression is:

$-12w^2 - 4x^2 + 2wx$

Example: Combining Like Terms in a Real-World Scenario

Suppose you are a physicist studying the motion of a particle. You have an expression that represents the kinetic energy of the particle:

$K = \frac{1}{2}mv^2 + \frac{1}{2}mv^2 + \frac{1}{2}mv^2 - \frac{1}{2}mv^2$

To simplify this expression, you need to combine the like terms. The like terms in this expression are:

• $\frac{1}{2}mv^2$ and $\frac{1}{2}mv^2$ (both have the variable $v$ raised to the power of 2)
• $\frac{1}{2}mv^2$ and $-\frac{1}{2}mv^2$ (both have the variable $v$ raised to the power of 2)

Combining the coefficients of the like terms, you get:

• $\frac{1}{2}mv^2$ and $\frac{1}{2}mv^2$: $\frac{1}{2} + \frac{1}{2} = 1$
• $\frac{1}{2}mv^2$ and $-\frac{1}{2}mv^2$: $\frac{1}{2} + (-\frac{1}{2}) = 0$

The simplified expression is:

$K = mv^2$

Conclusion

Combining like terms is an essential skill in algebra that helps simplify complex expressions. By following the step-by-step guide outlined in this article, you can combine like terms and simplify expressions in a variety of real-world scenarios. Whether you are a student, a teacher, or a professional, mastering the art of combining like terms will make you a more confident and effective problem solver.

Common Mistakes to Avoid

When combining like terms, it's essential to avoid common mistakes. Here are a few to watch out for:

• Not identifying like terms: Make sure to identify all the like terms in the expression before combining them.
• Not combining coefficients correctly: When combining coefficients, make sure to add or subtract them correctly.
• Not simplifying the expression: After combining like terms, make sure to simplify the expression by combining the combined terms.

Practice Problems

To practice combining like terms, try the following problems:

1. Combine the like terms in the expression: $3x^2 + 2x^2 - 4x^2 + 5x^2$
2. Combine the like terms in the expression: $2y^2 + 3y^2 - 2y^2 + 4y^2$
3. Combine the like terms in the expression: $-2z^2 + 3z^2 - 4z^2 + 5z^2$

Answer Key

1. $8x^2$
2. $6y^2$
3. $2z^2$

Final Thoughts

Frequently Asked Questions

Q: What are like terms in algebra?

A: Like terms are expressions that have the same variable(s) raised to the same power. They can have different coefficients, but the variable(s) and the exponent(s) must be the same.

Q: Why is it important to combine like terms?

A: Combining like terms is essential in algebra because it helps simplify complex expressions. By combining like terms, you can reduce the number of terms in an expression, making it more manageable and easier to work with.

Q: How do I identify like terms in an expression?

A: To identify like terms, look for terms that have the same variable(s) raised to the same power. For example, in the expression $2x^2 + 3x^2 - 4x^2$, the like terms are $2x^2$ and $3x^2$ because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, in the expression $2x^2 + 3x^2 - 4x^2$, the coefficients are 2, 3, and -4. Combining these coefficients, you get $2 + 3 - 4 = 1$. The combined term is $x^2$.

Q: What are some common mistakes to avoid when combining like terms?

A: Some common mistakes to avoid when combining like terms include:

• Not identifying like terms
• Not combining coefficients correctly
• Not simplifying the expression

Q: How do I simplify an expression after combining like terms?

A: To simplify an expression after combining like terms, combine the combined terms. For example, in the expression $2x^2 + 3x^2 - 4x^2$, the combined term is $x^2$. Simplifying the expression, you get $x^2$.

Q: Can I combine like terms with variables that have different exponents?

A: No, you cannot combine like terms with variables that have different exponents. For example, in the expression $2x^2 + 3x^3$, the terms are not like terms because they have different exponents.

Q: Can I combine like terms with variables that have different coefficients?

A: Yes, you can combine like terms with variables that have different coefficients. For example, in the expression $2x^2 + 3x^2$, the coefficients are 2 and 3. Combining these coefficients, you get $2 + 3 = 5$. The combined term is $5x^2$.

Q: How do I know if I have combined like terms correctly?

A: To check if you have combined like terms correctly, simplify the expression and see if it matches the original expression. If it does, then you have combined like terms correctly.

Q: Can I use a calculator to combine like terms?

A: Yes, you can use a calculator to combine like terms. However, it's always a good idea to check your work by hand to make sure you have combined like terms correctly.

Additional Resources

• Algebra textbooks: Check out algebra textbooks for more information on combining like terms.
• Online resources: Visit online resources such as Khan Academy, Mathway, and Wolfram Alpha for more information on combining like terms.
• Practice problems: Practice combining like terms with online practice problems or worksheets.

Conclusion

Combining like terms is an essential skill in algebra that helps simplify complex expressions. By following the step-by-step guide outlined in this article, you can combine like terms and simplify expressions in a variety of real-world scenarios. Remember to identify like terms, combine coefficients correctly, and simplify the expression to get the final answer. With practice and patience, you'll become a master of combining like terms in no time!