Solving Linear Equations: Calculate Expressions With A And B

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Hey guys! Today, we're diving into a fun math problem that involves solving linear equations. We've got two equations: 2a + b = 37 and 5a + 3b = 98. Our mission, should we choose to accept it (and we do!), is to calculate the values of a few expressions: 7a + 4b, 6a + 3b, 10a + 6b, and 16a + 9b. Sounds like a mathematical adventure, right? Let's put on our thinking caps and get started!

Understanding the Problem

Before we jump into solving, let's break down what we're dealing with. Linear equations are equations where the highest power of the variables (in this case, 'a' and 'b') is 1. We have a system of two linear equations, which means we have two equations with the same variables. To find the values of 'a' and 'b', we need to solve this system. Once we have 'a' and 'b', we can easily calculate the values of the expressions they ask us for. This involves using techniques like substitution or elimination to find the values of our variables. This type of problem is super common in algebra, and mastering it will help you tackle all sorts of math challenges down the road. Remember, practice makes perfect, so let's get into the nitty-gritty and show how it’s done.

Solving for 'a' and 'b'

Okay, let's roll up our sleeves and get into the heart of the problem: finding the values of 'a' and 'b'. We have two equations:

  1. 2a + b = 37
  2. 5a + 3b = 98

There are a couple of ways we can tackle this, but I'm going to use the elimination method because it's super effective. The goal here is to manipulate the equations so that either the 'a' terms or the 'b' terms cancel out when we add or subtract the equations. Looking at our equations, it seems easier to eliminate 'b'. To do this, we can multiply the first equation by -3. This gives us:

-6a - 3b = -111

Now we have:

  1. -6a - 3b = -111
  2. 5a + 3b = 98

See what's coming? We can add these two equations together, and the '-3b' and '+3b' will cancel each other out. So, let's do it:

(-6a - 3b) + (5a + 3b) = -111 + 98

This simplifies to:

-a = -13

So, a = 13. Awesome! We've found 'a'. Now we need to find 'b'. We can plug the value of 'a' back into either of our original equations. Let's use the first one, 2a + b = 37:

2(13) + b = 37

26 + b = 37

b = 37 - 26

b = 11

Fantastic! We've found that a = 13 and b = 11. Now that we have these values, we can move on to calculating those expressions they asked us about.

Calculating the Expressions

Now comes the fun part! We know a = 13 and b = 11, and we have four expressions to calculate. Let's take them one by one:

a) 7a + 4b

To calculate 7a + 4b, we simply substitute the values of 'a' and 'b':

7(13) + 4(11) = 91 + 44 = 135

So, 7a + 4b = 135. Easy peasy!

b) 6a + 3b

Next up, 6a + 3b. Again, we plug in our values:

6(13) + 3(11) = 78 + 33 = 111

Therefore, 6a + 3b = 111. We're on a roll!

c) 10a + 6b

Now let's tackle 10a + 6b:

10(13) + 6(11) = 130 + 66 = 196

So, 10a + 6b = 196. Looking good!

d) 16a + 9b

Last but not least, we have 16a + 9b:

16(13) + 9(11) = 208 + 99 = 307

And there we have it! 16a + 9b = 307. We've successfully calculated all four expressions. You see, once you nail down the values of 'a' and 'b', the rest is just straightforward arithmetic. Let’s take a moment to appreciate how we systematically broke down this problem and conquered it. Now, it’s time to recap our journey and solidify what we've learned.

Recap of Solutions

Alright, let's take a step back and recap what we've accomplished. We started with two equations, 2a + b = 37 and 5a + 3b = 98, and we were tasked with finding the values of four different expressions involving 'a' and 'b'. We used the elimination method to solve for 'a' and 'b', finding that a = 13 and b = 11. Then, we plugged these values into each expression:

  • 7a + 4b = 135
  • 6a + 3b = 111
  • 10a + 6b = 196
  • 16a + 9b = 307

So, there you have it! We've solved the entire problem step by step. This type of problem is a classic example of how to solve systems of linear equations and then use those solutions to evaluate other expressions. Now that we've nailed this one, let’s zoom out a bit and think about why these skills are so useful in the bigger picture of math and beyond. Understanding the broader applicability can really motivate us to keep learning and practicing.

Importance of Solving Linear Equations

So, why is all this equation-solving stuff so important? Well, solving linear equations is a fundamental skill in mathematics and has tons of real-world applications. Seriously, it's not just something they teach you in school to torture you! Think about it: linear equations can be used to model all sorts of situations, from calculating the cost of items at a store to figuring out the trajectory of a rocket. They're used in economics, engineering, computer science, and many other fields.

For example, in economics, linear equations can help model supply and demand curves. In engineering, they can be used to analyze circuits or design structures. In computer science, they're essential for creating algorithms and solving optimization problems. The ability to solve systems of equations allows you to analyze complex scenarios, make predictions, and optimize solutions. So, by mastering these skills, you're not just getting better at math—you're also preparing yourself for a wide range of opportunities in the future. Keep practicing, keep exploring, and you’ll find that math is an incredibly powerful tool!

Practice Problems

Now that we've worked through this problem together, it's your turn to shine! Practice is key to mastering any math skill, so let's try a few more problems. Here are a couple for you to try:

  1. If 3x + y = 25 and 2x + 2y = 30, calculate 5x + 3y.
  2. If 4p - q = 18 and p + q = 7, calculate 6p.

Work through these problems using the same techniques we used above. Remember to break them down step by step, and don't be afraid to make mistakes—that's how we learn! If you get stuck, go back and review our example, and think about the strategies we used. Solving equations can feel like a puzzle, and every puzzle you solve makes you a better problem-solver overall. So, grab a pencil and paper, and let's keep those math muscles flexing!