Equivalent Expressions: Find The Missing Value

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Equivalent Expressions: Find the Missing Value

Hey guys! Today, we're diving into the world of equivalent expressions and tackling a problem where we need to find a missing value. This is a super important concept in algebra, and mastering it will help you solve all sorts of equations and problems down the road. We'll break down the problem step-by-step, so you can understand exactly how to find the solution. Let's get started!

Understanding Equivalent Expressions

Before we jump into the problem, let's quickly review what equivalent expressions actually are. Think of it this way: equivalent expressions are like two different outfits that look different but ultimately have the same value. They might be written differently, but when you simplify them, they end up representing the same thing.

For example, 2 + 3 and 5 are equivalent expressions. Similarly, x + x and 2x are also equivalent. The key is that no matter what value you substitute for the variable (like x), both expressions will always give you the same result. This understanding is crucial when we are dealing with algebraic manipulations and solving for unknowns.

In our problem, we have two expressions: 6(3x + 5) and 15x + ? + 3x. Our goal is to find the value that we can put in the box (?) to make these two expressions equivalent. This involves a bit of algebraic simplification and careful comparison. Let’s delve deeper into how we can approach this task. Remember, patience and accuracy are your best friends in algebra. Let's make sure we apply the correct mathematical operations and follow the order of operations to get to the correct answer.

Problem Breakdown: 6(3x + 5) and 15x + ? + 3x

The question we're tackling today is: What value makes the expressions 6(3x + 5) and 15x + ? + 3x equivalent? We have a multiple-choice question with the following options:

A. 5 B. 11 C. 15 D. 30

To solve this, we need to figure out what number should replace the question mark to make the two expressions equal. The first expression is 6(3x + 5). This means we need to distribute the 6 across the terms inside the parentheses. Remember the distributive property? It says that a(b + c) = ab + ac. Applying this property here, we get:

6 * 3x + 6 * 5 = 18x + 30

So, the first expression simplifies to 18x + 30. Now let's look at the second expression: 15x + ? + 3x. We can combine the like terms 15x and 3x:

15x + 3x = 18x

This simplifies our second expression to 18x + ?. Now, we have two simplified expressions: 18x + 30 and 18x + ?. For these two expressions to be equivalent, the missing value (?) must be equal to 30. Therefore, the correct answer is D. 30. Remember, the key here was to simplify each expression as much as possible and then compare them. This is a common strategy in algebra, and it will serve you well in many different types of problems.

Step-by-Step Solution

Let's walk through the solution step-by-step to make sure we've got it crystal clear. This will reinforce the process and help you tackle similar problems in the future.

  1. Simplify the first expression: We start with 6(3x + 5). To simplify, we use the distributive property, which means we multiply the 6 by both terms inside the parentheses:

    6 * 3x = 18x 6 * 5 = 30

    So, 6(3x + 5) becomes 18x + 30.

  2. Simplify the second expression: The second expression is 15x + ? + 3x. We can combine the like terms, which are the terms with x in them:

    15x + 3x = 18x

    This simplifies the second expression to 18x + ?

  3. Compare the expressions: Now we have two simplified expressions: 18x + 30 and 18x + ?. For these expressions to be equivalent, they must be equal for all values of x. Notice that both expressions have the term 18x. The only difference is the constant term.

  4. Find the missing value: To make the expressions equal, the missing value (?) must be the same as the constant term in the first expression, which is 30.

Therefore, the value that makes the expressions equivalent is 30. The correct answer is D. 30. Remember, the key to solving these problems is to simplify each expression as much as possible and then compare the simplified forms. This approach makes it much easier to see what's missing and find the solution.

Why Other Options are Incorrect

Understanding why the incorrect options are wrong is just as important as knowing why the correct answer is right. It helps you solidify your understanding of the concepts and avoid common mistakes. Let's take a look at why options A, B, and C are incorrect in this case.

  • A. 5: If we substitute 5 for the question mark, the second expression becomes 15x + 5 + 3x, which simplifies to 18x + 5. This is not equivalent to 18x + 30, because the constant terms are different. So, 5 is not the correct value.
  • B. 11: Similarly, if we use 11 as the missing value, the second expression becomes 15x + 11 + 3x, which simplifies to 18x + 11. Again, this is not equivalent to 18x + 30, because the constant terms don't match. Therefore, 11 is incorrect.
  • C. 15: If we try 15 as the missing value, the second expression becomes 15x + 15 + 3x, which simplifies to 18x + 15. This is also not equivalent to 18x + 30, as the constant terms are different. Hence, 15 is not the correct value.

By understanding why these options are wrong, you can see how crucial it is to correctly simplify and compare the expressions. The only way for the two expressions to be equivalent is if they are exactly the same when simplified. This means that all corresponding terms must be equal.

Tips for Solving Similar Problems

Now that we've solved this problem, let's talk about some tips that can help you tackle similar questions in the future. These strategies will make your problem-solving process more efficient and accurate.

  1. Always simplify first: The first and most important step is to simplify each expression as much as possible. This usually involves using the distributive property, combining like terms, and performing any other necessary algebraic operations. Simplifying makes it much easier to compare expressions and identify missing values.
  2. Pay attention to the distributive property: The distributive property is a key tool for simplifying expressions with parentheses. Remember to multiply the term outside the parentheses by every term inside the parentheses.
  3. Combine like terms: Like terms are terms that have the same variable raised to the same power. Combining like terms helps to simplify expressions and make them easier to compare.
  4. Compare corresponding parts: Once you've simplified the expressions, compare the corresponding parts (e.g., the terms with x, the constant terms). This will help you identify what's missing and what value needs to be inserted to make the expressions equivalent.
  5. Check your answer: After you've found a solution, plug it back into the original expressions to make sure they are indeed equivalent. This is a great way to catch any errors and ensure that your answer is correct. Remember, double-checking your work is always a good habit in mathematics.

By following these tips, you'll be well-equipped to solve a wide range of problems involving equivalent expressions. Practice makes perfect, so keep working on these types of questions to build your skills and confidence.

Practice Makes Perfect

The best way to master equivalent expressions is through practice. The more problems you solve, the more comfortable you'll become with the techniques and strategies involved. So, let's explore some additional practice problems that are similar to the one we just solved. Remember, the key is to simplify, compare, and conquer!

For example, you might encounter a problem like this: What value makes the expressions 4(2x - 3) and 5x + ? - x equivalent? To solve this, you would follow the same steps we outlined earlier:

  1. Simplify 4(2x - 3) using the distributive property.
  2. Simplify 5x + ? - x by combining like terms.
  3. Compare the simplified expressions and find the missing value.
  4. Check your answer by substituting it back into the original expressions.

Another type of problem you might encounter is finding the value of a variable that makes two expressions equivalent for a specific value of x. For example: Find the value of k that makes the expressions 3x + k and 7x - 4 equivalent when x = 2. In this case, you would substitute x = 2 into both expressions, set them equal to each other, and solve for k.

Remember, each problem is a chance to sharpen your skills and deepen your understanding. So, don't be afraid to tackle challenging questions and learn from your mistakes. Keep practicing, and you'll become a pro at working with equivalent expressions!

Conclusion

Alright, guys, we've covered a lot today about equivalent expressions and how to find missing values! We've learned that equivalent expressions are like different outfits that have the same value underneath. We've broken down the steps to solve problems like this, from simplifying expressions using the distributive property and combining like terms, to comparing the simplified forms and finding the missing piece of the puzzle.

We also emphasized the importance of understanding why the incorrect options are wrong, which helps solidify your grasp of the concepts. And, of course, we talked about some awesome tips and tricks to help you tackle similar problems in the future, like always simplifying first and double-checking your answers.

Remember, practice is key! The more you work with these types of problems, the more confident and skilled you'll become. So, keep practicing, and don't hesitate to ask for help if you get stuck. You've got this! Keep up the great work, and we'll see you next time with more math adventures!