Math Problem: Solving Expressions With Given Variables

Alright, guys, let's dive into this math problem where we need to calculate the values of several expressions given that a + b + c = 23 and x = 9. This type of problem is all about using the distributive property and substitution to simplify things. We'll break down each part step by step so it’s super clear. Let's get started!

a) 14a + 14b + 14c + 14x

In this first expression, we've got 14a + 14b + 14c + 14x. The key here is to recognize that 14 is a common factor in the first three terms. We can use the distributive property to factor out the 14. So, let’s rewrite the expression:

14(a + b + c) + 14x

Now, remember our given information? We know that a + b + c = 23 and x = 9. We can now substitute these values into our expression:

14(23) + 14(9)

Let's calculate each part:

14 * 23 = 322 14 * 9 = 126

Now, add those results together:

322 + 126 = 448

So, the value of the expression 14a + 14b + 14c + 14x is 448. See how breaking it down and using the distributive property makes it much easier to handle? We took a seemingly complex expression and turned it into simple arithmetic. This is a core strategy in solving algebraic problems, and mastering it can make those tricky equations a whole lot less intimidating.

b) 2013 - (53a + 53b + 53c + 10x)

For the second expression, we have 2013 - (53a + 53b + 53c + 10x). Again, the trick here is to identify common factors and use substitution. Notice that 53 is a common factor in the first three terms inside the parentheses. Let's factor it out:

2013 - (53(a + b + c) + 10x)

Now, just like before, we substitute the values we know: a + b + c = 23 and x = 9:

2013 - (53(23) + 10(9))

First, we need to calculate the values inside the parentheses:

53 * 23 = 1219 10 * 9 = 90

Now, add those together:

1219 + 90 = 1309

So, our expression now looks like this:

2013 - 1309

Finally, subtract:

2013 - 1309 = 704

Therefore, the value of the expression 2013 - (53a + 53b + 53c + 10x) is 704. By carefully factoring and substituting, we avoided a lot of messy calculations and arrived at the solution smoothly. This approach is super helpful in managing larger expressions and reducing the chances of making errors.

c) 349 + 6a + 6b + 6c - 12x

Moving on to our third expression, 349 + 6a + 6b + 6c - 12x, let’s continue using the same strategies. We spot that 6 is a common factor in the 6a, 6b, and 6c terms. So, we'll factor that out:

349 + 6(a + b + c) - 12x

Now, let's substitute our known values a + b + c = 23 and x = 9:

349 + 6(23) - 12(9)

Time to do the calculations:

6 * 23 = 138 12 * 9 = 108

Plug those back into the expression:

349 + 138 - 108

Now, add and subtract:

349 + 138 = 487 487 - 108 = 379

So, the value of the expression 349 + 6a + 6b + 6c - 12x is 379. By consistently applying factoring and substitution, we're making these problems much more manageable. It’s like having a set of tools that we can use to disassemble a complex problem into smaller, easier pieces. This is a skill that will be super useful in more advanced math topics too!

d) 424 - 21x + 3a + 3b + 3c

Okay, let's tackle the fourth expression: 424 - 21x + 3a + 3b + 3c. Once again, we look for common factors. We see that 3 is common in the terms 3a, 3b, and 3c. Let’s factor it out:

424 - 21x + 3(a + b + c)

Now, substitute the values a + b + c = 23 and x = 9:

424 - 21(9) + 3(23)

Let’s do the calculations:

21 * 9 = 189 3 * 23 = 69

Substitute these back into the expression:

424 - 189 + 69

Now, let's do the addition and subtraction:

424 - 189 = 235 235 + 69 = 304

So, the value of the expression 424 - 21x + 3a + 3b + 3c is 304. This method of breaking down the problem—factoring and then substituting—really helps in keeping the calculations organized and accurate. You guys are doing great! Keep practicing this, and you’ll find these types of problems become second nature.

e) 7a + 7b + 7c - 10x

Finally, let's handle the last expression: 7a + 7b + 7c - 10x. Just like before, we look for common factors. Here, 7 is common in the terms 7a, 7b, and 7c. So, we factor it out:

7(a + b + c) - 10x

Now, we substitute our known values a + b + c = 23 and x = 9:

7(23) - 10(9)

Time for the calculations:

7 * 23 = 161 10 * 9 = 90

Substitute those back into the expression:

161 - 90

Finally, subtract:

161 - 90 = 71

Thus, the value of the expression 7a + 7b + 7c - 10x is 71. You see, by consistently using the same approach—identifying common factors, factoring them out, substituting known values, and then doing the arithmetic—we can solve these expressions quite efficiently. This consistent method not only simplifies the process but also helps prevent errors.

Conclusion

So, guys, we've worked through each of these expressions step by step, and hopefully, you're feeling more confident about tackling similar problems. The key takeaways here are the importance of identifying common factors, using the distributive property to simplify, and then substituting the given values. These techniques are fundamental in algebra and will serve you well as you continue your math journey. Remember, practice makes perfect, so keep at it, and you’ll become a pro at solving these types of problems in no time!