Math Truth Test: Cracking The Code Of Equations

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Hey guys, let's dive into some math problems! Today, we're going to examine several equations and figure out if they're true or false. It's like being a math detective, except instead of finding clues, we're checking if the numbers add up correctly. We will explore each statement, breaking down the steps and making it super easy to understand. Let's get started and see if we can ace this math truth test! So, grab your pencils, and let's get our math on.

Decoding the Truth: Statement (a) - 33 * 29 + 33 * 74 = 33(29 + 74)

Alright, let's start with statement (a): 33 * 29 + 33 * 74 = 33(29 + 74). What we're dealing with here is a fundamental concept in math known as the distributive property. Essentially, the distributive property says that when you have a number multiplied by a sum inside parentheses, you can distribute that number to each term within the parentheses. In other words, a(b + c) = ab + ac. In this specific equation, it is checking our understanding of the distributive property of multiplication over addition. This property lets us simplify complex calculations by breaking them down into smaller, more manageable steps. It's a handy tool that shows up all over the place in algebra and beyond, making it really important to get a good grasp of it.

So, let's simplify the left side of the equation. First, we need to perform the multiplications: 33 multiplied by 29, and 33 multiplied by 74. Then, you add the results. Next, we need to look at the right side of the equation. Here, we first add 29 and 74, and then multiply the result by 33. The goal is to see if the left side equals the right side. If the results match, the statement is true. If they do not, it is false. Keep in mind that we're basically checking to see if we have applied the distributive property correctly. If both sides of the equation are equal, it means that the distributive property has been used correctly. By working through the equation step by step, we will be able to prove if the original statement is correct or incorrect. It’s all about following the order of operations and understanding how the different parts of the equation relate to each other. By getting the answer correctly, you can understand how these simple rules can apply to the more complex equations that you might encounter later on. The distributive property will be one of the critical concepts of mathematical proficiency.

Let’s start doing the math. 33 * 29 = 957, and 33 * 74 = 2442. Adding these two results, 957 + 2442 = 3399. Now let's calculate the right side: 29 + 74 = 103, and 33 * 103 = 3399. Both sides equal the same number, so this means that the statement is true!

Unraveling Statement (b): 58 * 92 - 25 * 58 = 58(92 - 25)

Now, let's shift gears and focus on statement (b): 58 * 92 - 25 * 58 = 58(92 - 25). This one is similar to the first statement, but it now involves subtraction. Again, we will be using the distributive property, but this time, it's the distributive property of multiplication over subtraction: a(b - c) = ab - ac. The distributive property provides a convenient way to simplify expressions. Just like before, we'll examine both sides of the equation to see if they are equal. The goal is to prove whether the statement is true or false. We'll follow the same procedure to check if the distributive property is applied correctly. If the results are equal, it confirms the accuracy of the statement. If not, the statement is proven to be false.

Let's start by calculating the left side. First, we need to perform the multiplications: 58 multiplied by 92 and 25 multiplied by 58. Then subtract the results. On the right side, we first subtract 25 from 92 and then multiply the result by 58. Keep in mind, the order of operations is crucial here, and each step has to be calculated properly.

Doing the calculations: 58 * 92 = 5336 and 25 * 58 = 1450. Then subtract 1450 from 5336 and we get 3886. Next, calculate the right side: 92 - 25 = 67, and 58 * 67 = 3886. The results from both sides are the same, which means the equation is true!

Dissecting Statement (c): 61 * 123 + 123 * 35 = 123(61 + 35)

Let's keep the math train rolling and analyze statement (c): 61 * 123 + 123 * 35 = 123(61 + 35). This statement reinforces our understanding of the distributive property, specifically the one over addition, and provides another opportunity to test our calculation skills. We'll walk through the process to simplify each side of the equation and compare the final results.

For the left side of the equation, we need to perform two multiplications first: 61 multiplied by 123, and 123 multiplied by 35. After these, you then add the products together. For the right side of the equation, add 61 and 35, and then multiply the result by 123.

Let’s do the math. 61 * 123 = 7463, and 123 * 35 = 4305. Adding these two results, 7463 + 4305 = 11768. Now calculate the right side: 61 + 35 = 96, and 123 * 96 = 11808. In this case, both sides are not equal, so the statement is false!

Unveiling Statement (d): 89 * 508 - 508 * 45 = 508(89 - 45)

Finally, let's explore statement (d): 89 * 508 - 508 * 45 = 508(89 - 45). This statement provides another opportunity to practice our skills of applying the distributive property of multiplication over subtraction and shows how mathematical concepts come into play. We're going to break down the calculations step by step to determine if this statement holds true.

On the left side, we start by calculating the multiplications: 89 multiplied by 508, and 45 multiplied by 508. After finding the individual products, subtract the second result from the first. Then, for the right side, subtract 45 from 89, then multiply the difference by 508.

Let’s proceed with the calculations. 89 * 508 = 45232, and 45 * 508 = 22860. Then, 45232 - 22860 = 22372. Now, on the right side: 89 - 45 = 44, and 44 * 508 = 22352. The results are not equal, which means that the statement is false!

Conclusion: Truth or Dare in Math!

So, after working through all the statements, we've completed our math truth test! Remember, guys, the distributive property is your friend. Keep practicing, and you'll become pros at these types of problems.

Here’s a summary of our findings:

  • Statement (a): True
  • Statement (b): True
  • Statement (c): False
  • Statement (d): False

Keep up the great work and keep exploring the amazing world of math!