Combining Like Terms: Step-by-Step Guide

Combining Like Terms: Solving $5(x-3)+7x-8$ (With Step-by-Step Solution)

Hey guys! Let's dive into a common algebra problem: combining like terms. This is a fundamental skill, and we'll break down how to simplify the expression $5(x-3)+7x-8$. We'll go through the steps, explain the logic, and then see which of the multiple-choice options is the correct answer. Get ready to flex those math muscles!

Understanding the Problem: Combining Like Terms

Okay, so what does "combining like terms" actually mean? Well, it's all about simplifying an algebraic expression by grouping together terms that are similar. Think of it like sorting your toys: you put all the cars together, all the action figures together, and so on. In algebra, "like terms" are terms that have the same variable raised to the same power. For example, $7x$ and $2x$ are like terms because they both have the variable $x$ raised to the power of 1. On the other hand, $x^2$ and $x$ are not like terms because they have different powers.

Our expression, $5(x-3)+7x-8$, has a few parts we need to deal with. We've got parentheses, multiplication, terms with $x$, and constant terms (numbers without variables). The key is to follow the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to simplify it step by step. Remember, the goal is to get the expression into a simpler form where we have fewer terms.

Before we get started, it's important to understand the concept of the distributive property. This property allows us to multiply a term outside the parentheses by each term inside the parentheses. It's a crucial step in simplifying expressions like ours. The distributive property states that $a(b + c) = ab + ac$. We will use this property to eliminate the parenthesis and combine the like terms. Keep in mind that practice makes perfect. The more you practice, the more comfortable you'll become with these types of problems. This problem is a great example of how important it is to understand and apply the distributive property, which makes it a great starting point for anyone trying to master algebra. So let's get into it and break down this question!

Step-by-Step Solution: Breaking Down the Expression

Alright, let's get to work! We'll simplify $5(x-3)+7x-8$ step by step.

1. Distribute the 5: First, we need to get rid of those parentheses. We do this by distributing the 5 to both terms inside the parentheses: $5 * x$ and $5 * -3$. This gives us $5x - 15$.

2. Rewrite the Expression: Now our expression looks like this: $5x - 15 + 7x - 8$.

3. Identify Like Terms: Now, let's identify the like terms. We have two terms with $x$: $5x$ and $7x$, and we have two constant terms: $-15$ and $-8$.

4. Combine Like Terms: Let's combine the $x$ terms: $5x + 7x = 12x$. Then, combine the constant terms: $-15 - 8 = -23$.

5. Final Simplified Expression: Putting it all together, our simplified expression is $12x - 23$.

So, we've successfully combined like terms and simplified the expression. We've taken a complex expression and turned it into a much cleaner, more manageable form. This is super important for solving equations and understanding more complex algebra concepts later on. Understanding how to do these types of operations will make your life a lot easier as you go through algebra.

Choosing the Correct Answer: Multiple Choice Options

Now that we've done the work, let's look at the multiple-choice options provided and see which one matches our answer. We have:

• (a) $2x - 5$
• (b) $12x - 23$
• (c) $12x - 11$
• (d) $-3x + 5$

Our simplified expression is $12x - 23$. Comparing this to the options, we can see that option (b) is the correct answer. Bingo! We've successfully solved the problem and identified the correct multiple-choice answer. Knowing how to do this problem is like having a superpower. It allows you to tackle a bunch of algebra questions and other related topics. Keep in mind that practice is key to mastering any math concept, and combining like terms is no exception. The more problems you work through, the more confident you'll become in your abilities. So keep at it, and you'll be acing those algebra tests in no time!

Conclusion: Mastering Combining Like Terms

Congratulations, guys! We've successfully navigated the process of combining like terms in the expression $5(x-3)+7x-8$. We've seen how to distribute, identify like terms, combine them, and arrive at the simplified form: $12x - 23$. Remember, this is a fundamental skill in algebra, and it will serve you well as you tackle more complex problems.

Key Takeaways:

• Distributive Property: Always use the distributive property to remove parentheses.
• Identify Like Terms: Carefully identify terms with the same variable and power, and the constant terms.
• Combine Like Terms: Add or subtract the coefficients of the like terms.
• Simplify: Always simplify your answer to its simplest form.

Keep practicing, and you'll become a pro at combining like terms in no time! Keep up the great work and enjoy the process. The most important thing is to practice, and you will get better! This step-by-step guide should help you understand the concepts in more depth and will make you well on your way to mastering this skill!

Further Practice and Resources

Want to level up your skills? Here are some ideas:

• Practice Problems: Search online for "combining like terms worksheets" or "algebra practice problems." Work through various examples to solidify your understanding.
• Online Tutorials: Check out websites like Khan Academy or YouTube for video tutorials on combining like terms. Visual explanations can be incredibly helpful.
• Seek Help: Don't be afraid to ask your teacher, classmates, or a tutor for help if you're struggling. Getting help is a sign of strength, not weakness.

Keep practicing, stay curious, and keep exploring the exciting world of algebra! You've got this!