Simplify By Combining Like Terms: \[$-15 + 6x - 19 - 20x\$\]

Feb 26, 2025 by ADMIN 61 views

**Simplify by Combining Like Terms: A Comprehensive Guide**

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 explore how to simplify the expression ${$ -15 + 6x - 19 - 20x$}$ by combining like terms.

What are Like Terms?

Like terms are terms that have the same variable and exponent. For example, ${ $2x\$}$ and ${ $3x\$}$ are like terms because they both have the variable ${$ x$}$ and the same exponent, which is 1. On the other hand, ${ $2x\$}$ and ${ $3y\$}$ are not like terms because they have different variables.

Step 1: Identify the Like Terms

To simplify the expression ${$ -15 + 6x - 19 - 20x$}$, we need to identify the like terms. In this case, the like terms are the constant terms ${$ -15$}$ and ${$ -19$}$, and the terms with the variable ${$ x$}$, which are ${ $6x\$}$ and ${$ -20x$}$.

Step 2: Combine the Constant Terms

The constant terms are the terms that do not have any variable. In this case, the constant terms are ${$ -15$}$ and ${$ -19$}$. To combine these terms, we add their coefficients, which are the numbers in front of the terms. So, ${$ -15 + (-19)$] = [ $-34\$}$ .

Step 3: Combine the Like Terms with the Variable

The like terms with the variable are ${ $6x\$}$ and ${$ -20x$}$. To combine these terms, we add their coefficients, which are the numbers in front of the terms. So, ${ $6x + (-20x)\$] = \[$ -14x$}$.

Step 4: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by combining the constant term and the term with the variable. So, the simplified expression is ${$ -34 - 14x$}$.

Example

Let's consider another example to illustrate the concept of combining like terms. Suppose we have the expression ${ $2x + 3y - 4x + 2y\$}$ . To simplify this expression, we need to identify the like terms, which are the terms with the same variable. In this case, the like terms are ${ $2x\$}$ and ${$ -4x$}$, and the terms with the variable ${$ y$}$, which are ${ $3y\$}$ and ${ $2y\$}$ . We can then combine these terms by adding their coefficients. So, ${ $2x + (-4x)\$] = \[$ -2x$}$, and ${ $3y + 2y}$ $] = ${ $5y\$}$ . The simplified expression is ${$ -2x + 5y$}$.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions by combining like terms:

Identify the like terms: The first step in simplifying an expression is to identify the like terms. Look for terms that have the same variable and exponent.
Combine the constant terms: Combine the constant terms by adding their coefficients.
Combine the like terms with the variable: Combine the like terms with the variable by adding their coefficients.
Simplify the expression: Combine the constant term and the term with the variable to simplify the expression.

Conclusion

Combining like terms is a fundamental concept in algebra that helps simplify complex expressions. By identifying the like terms, combining the constant terms, combining the like terms with the variable, and simplifying the expression, we can simplify expressions and make them easier to work with. With practice and patience, you can become proficient in combining like terms and simplify expressions with ease.

Common Mistakes to Avoid

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

Not identifying the like terms: Failing to identify the like terms can lead to incorrect simplification of the expression.
Not combining the constant terms: Failing to combine the constant terms can lead to incorrect simplification of the expression.
Not combining the like terms with the variable: Failing to combine the like terms with the variable can lead to incorrect simplification of the expression.
Not simplifying the expression: Failing to simplify the expression can lead to incorrect solutions to algebraic problems.

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 is used to simplify complex equations that describe the motion of objects. In engineering, combining like terms is used to simplify complex equations that describe the behavior of electrical circuits. In economics, combining like terms is used to simplify complex equations that describe the behavior of economic systems.

Final Thoughts

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, ${ $2x\$}$ and ${ $3x\$}$ are like terms because they both have the variable ${$ x$}$ and the same exponent, which is 1.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable and exponent. For example, in the expression ${ $2x + 3y - 4x + 2y\$}$ , the like terms are ${ $2x\$}$ and ${$ -4x$}$, and the terms with the variable ${$ y$}$, which are ${ $3y\$}$ and ${ $2y\$}$ .

Q: How do I combine like terms?

A: To combine like terms, add their coefficients. For example, in the expression ${ $2x + 3y - 4x + 2y\$}$ , the like terms are ${ $2x\$}$ and ${$ -4x$}$, and the terms with the variable ${$ y$}$, which are ${ $3y\$}$ and ${ $2y\$}$ . We can then combine these terms by adding their coefficients. So, ${ $2x + (-4x)\$] = \[$ -2x$}$, and ${ $3y + 2y}$ $] = ${ $5y\$}$ .

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms is a step in simplifying an expression. Simplifying an expression involves combining like terms, as well as eliminating any unnecessary terms or operations.

Q: Can I combine unlike terms?

A: No, unlike terms cannot be combined. Unlike terms are terms that have different variables or exponents. For example, ${ $2x\$}$ and ${ $3y\$}$ are unlike terms because they have different variables.

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 the like terms
Not combining the constant terms
Not combining the like terms with the variable
Not simplifying the expression

Q: How do I know when to combine like terms?

A: You should combine like terms whenever you have an expression that contains multiple terms with the same variable and exponent. Combining like terms can help simplify the expression and make it easier to work with.

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're getting the correct answer.

Q: What are some real-world applications of combining like terms?

A: Combining like terms has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, combining like terms is used to simplify complex equations that describe the motion of objects. In engineering, combining like terms is used to simplify complex equations that describe the behavior of electrical circuits. In economics, combining like terms is used to simplify complex equations that describe the behavior of economic systems.

Q: How do I practice combining like terms?

A: You can practice combining like terms by working through algebraic expressions and simplifying them. You can also try combining like terms with different variables and exponents to get a feel for how it works.

Q: What are some tips for mastering combining like terms?

A: Some tips for mastering combining like terms include:

Practice, practice, practice: The more you practice combining like terms, the more comfortable you'll become with the process.
Start with simple expressions: Begin with simple expressions and gradually work your way up to more complex ones.
Use a calculator: If you're having trouble combining like terms by hand, try using a calculator to check your work.
Check your work: Always check your work to make sure you're getting the correct answer.