Combine Like Terms In The Expression: ${ 7x^2 - 2x + 8x^2 + 3x }$

Introduction

In algebra, combining like terms is a fundamental concept that helps simplify complex expressions. It involves adding or subtracting terms that have the same variable and exponent. In this article, we will focus on combining like terms in the given expression: $7x^2 - 2x + 8x^2 + 3x$. We will break down the process step by step and provide examples to illustrate the concept.

What are Like Terms?

Like terms are terms that have the same variable and exponent. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ and the exponent $1$. Similarly, $3x^2$ and $7x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Step 1: Identify Like Terms

To combine like terms, we need to identify the terms that have the same variable and exponent. In the given expression, we can see that there are two terms with the variable $x^2$: $7x^2$ and $8x^2$. We can also see that there are two terms with the variable $x$: $-2x$ and $3x$.

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them. To combine like terms, we add or subtract the coefficients of the terms. In this case, we can combine the two terms with the variable $x^2$ by adding their coefficients: $7x^2 + 8x^2 = 15x^2$. We can also combine the two terms with the variable $x$ by adding their coefficients: $-2x + 3x = x$.

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by combining the resulting terms. In this case, we can combine the two terms: $15x^2 + x$.

Example

Let's consider another example to illustrate the concept of combining like terms. Suppose we have the expression: $4x^2 + 2x - 3x^2 + 5x$. We can identify the like terms as $4x^2$ and $-3x^2$ (both have the variable $x^2$) and $2x$ and $5x$ (both have the variable $x$). We can combine the like terms by adding or subtracting their coefficients: $4x^2 - 3x^2 = x^2$ and $2x + 5x = 7x$. The simplified expression is: $x^2 + 7x$.

Tips and Tricks

Here are some tips and tricks to help you combine like terms:

• Pay attention to the signs: When combining like terms, make sure to pay attention to the signs of the coefficients. If the signs are the same, add the coefficients. If the signs are different, subtract the coefficients.
• Use parentheses: To avoid confusion, use parentheses to group like terms together.
• Check your work: Once you have combined the like terms, check your work by plugging in values for the variable.

Conclusion

Combining like terms is a fundamental concept in algebra that helps simplify complex expressions. By identifying like terms, combining them, and simplifying the expression, we can make complex expressions more manageable. Remember to pay attention to the signs, use parentheses, and check your work to ensure accuracy.

Common Mistakes to Avoid

Here are some common mistakes to avoid when combining like terms:

• Not identifying like terms: Make sure to identify like terms carefully to avoid combining terms that are not alike.
• Not paying attention to signs: Pay attention to the signs of the coefficients to avoid making mistakes when combining like terms.
• Not using parentheses: Use parentheses to group like terms together to avoid confusion.

Real-World Applications

Combining like terms has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, combining like terms can help simplify complex equations that describe the motion of objects. In engineering, combining like terms can help design more efficient systems. In economics, combining like terms can help analyze complex data sets.

Practice Problems

Here are some practice problems to help you practice combining like terms:

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

Answer Key

Here are the answers to the practice problems:

• $3x^2 + 2x - 4x^2 + 5x = -x^2 + 7x$
• $2x^2 - 3x + 5x^2 - 2x = 7x^2 - 5x$
• $4x^2 + 3x - 2x^2 - 5x = 2x^2 - 2x$

Conclusion

Introduction

Combining like terms is a fundamental concept in algebra that helps simplify complex expressions. In our previous article, we discussed the basics of combining like terms and provided examples to illustrate the concept. In this article, we will answer some frequently asked questions about combining like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ and the exponent $1$. Similarly, $3x^2$ and $7x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable and exponent. You can also use the following steps:

1. Write down the terms in the expression.
2. Look for terms that have the same variable and exponent.
3. Group the like terms together.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the terms. For example, if you have the expression $3x^2 + 2x^2$, you can combine the like terms by adding the coefficients: $3x^2 + 2x^2 = 5x^2$.

Q: What if I have a negative coefficient?

A: If you have a negative coefficient, you can combine it with other like terms by subtracting the coefficients. For example, if you have the expression $-2x^2 + 3x^2$, you can combine the like terms by subtracting the coefficients: $-2x^2 + 3x^2 = x^2$.

Q: Can I combine like terms with different variables?

A: No, you cannot combine like terms with different variables. For example, you cannot combine the terms $2x$ and $3y$ because they have different variables.

Q: Can I combine like terms with different exponents?

A: No, you cannot combine like terms with different exponents. For example, you cannot combine the terms $2x^2$ and $3x$ because they have different exponents.

Q: How do I simplify an expression with like terms?

A: To simplify an expression with like terms, combine the like terms and then simplify the resulting expression. For example, if you have the expression $3x^2 + 2x^2 + 5x$, you can combine the like terms by adding the coefficients: $3x^2 + 2x^2 = 5x^2$. Then, you can simplify the resulting expression by combining the like terms: $5x^2 + 5x = 5x^2 + 5x$.

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 get the correct answer.

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 carefully
• Not paying attention to the signs of the coefficients
• Not using parentheses to group like terms together
• Not checking your work by hand

Conclusion

Combining like terms is a fundamental concept in algebra that helps simplify complex expressions. By identifying like terms, combining them, and simplifying the resulting expression, you can make complex expressions more manageable. Remember to pay attention to the signs, use parentheses, and check your work to ensure accuracy. With practice, you will become more comfortable combining like terms and be able to apply this concept to real-world problems.

Practice Problems

Here are some practice problems to help you practice combining like terms:

• Combine the like terms in the expression: $3x^2 + 2x^2 - 4x^2 + 5x$
• Combine the like terms in the expression: $2x^2 - 3x + 5x^2 - 2x$
• Combine the like terms in the expression: $4x^2 + 3x - 2x^2 - 5x$

Answer Key

Here are the answers to the practice problems:

• $3x^2 + 2x^2 - 4x^2 + 5x = -x^2 + 7x$
• $2x^2 - 3x + 5x^2 - 2x = 7x^2 - 5x$
• $4x^2 + 3x - 2x^2 - 5x = 2x^2 - 2x$

Real-World Applications

Combining like terms has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, combining like terms can help simplify complex equations that describe the motion of objects. In engineering, combining like terms can help design more efficient systems. In economics, combining like terms can help analyze complex data sets.

Conclusion

Combining like terms is a fundamental concept in algebra that helps simplify complex expressions. By identifying like terms, combining them, and simplifying the resulting expression, you can make complex expressions more manageable. Remember to pay attention to the signs, use parentheses, and check your work to ensure accuracy. With practice, you will become more comfortable combining like terms and be able to apply this concept to real-world problems.