Solving Systems Of Equations Finding The Value Of 5x + 7y

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Hey guys! Math can sometimes feel like navigating a maze, especially when you're faced with systems of equations. But don't worry, we're here to break it down and make it super easy to understand. Today, we're tackling a classic problem where we need to find the value of an expression given a system of equations. So, let's dive right into it and solve this math puzzle together!

The Problem

Okay, so here's the problem we're going to solve. We've got two equations:

  1. 2x + 5y = 19
  2. 11x + 5y = 37

And the big question? What is the value of 5x + 7y? Sounds a bit tricky, right? But trust me, with a systematic approach, it's totally manageable. We're going to walk through it step by step, so you'll not only get the answer but also understand the method behind it. Understanding the method is the real key here, because once you get that, you can apply it to all sorts of similar problems. Think of it as learning a new superpower – the ability to crack any system of equations that comes your way!

Understanding Systems of Equations

Before we jump into the solution, let's quickly recap what a system of equations actually is. A system of equations is simply a set of two or more equations that share the same variables. In our case, we have two equations, both involving the variables x and y. The goal is to find values for these variables that satisfy both equations simultaneously. Think of it like finding the perfect combination that unlocks both equations. There are several methods to solve systems of equations, and we'll be using a common one called elimination. Elimination is a fantastic method because it allows us to get rid of one variable, making it easier to solve for the other. It's like strategically removing obstacles one by one until you reach your goal. This method is particularly useful when the coefficients (the numbers in front of the variables) of one of the variables are the same or can be easily made the same. So, keep this in mind as we move forward, because elimination is going to be our trusty tool in this math adventure!

Step 1 Eliminate One Variable

Our first move is to eliminate one of the variables. Looking at our equations:

  1. 2x + 5y = 19
  2. 11x + 5y = 37

Notice anything? The '5y' term is the same in both equations! This is perfect for the elimination method. To eliminate 'y', we'll subtract the first equation from the second equation. Why subtract? Because 5y - 5y equals zero, effectively removing 'y' from the equation. This is like performing a mathematical magic trick where we make a variable disappear! When we subtract equations, it's crucial to subtract each term correctly. We'll subtract the left-hand sides and the right-hand sides separately. This ensures we maintain the balance of the equation. It's like a seesaw – whatever you do on one side, you have to do on the other to keep it level. So, let's get to the subtraction and see what simplified equation we get.

Performing the Subtraction

Let's perform the subtraction carefully. We'll take equation (2) and subtract equation (1) from it:

(11x + 5y) - (2x + 5y) = 37 - 19

Now, let's break this down step by step. First, we'll subtract the 'x' terms: 11x - 2x = 9x. Next, we subtract the 'y' terms: 5y - 5y = 0. As planned, the 'y' variable is eliminated! Finally, we subtract the constants: 37 - 19 = 18. So, after the subtraction, our equation looks like this:

9x = 18

Isn't that neat? We've transformed two complex equations into a single, simple equation with just one variable. This is a huge step forward! Now, we have a much clearer path to finding the value of 'x'. All that's left to do is solve this straightforward equation. We're on the home stretch for finding 'x', so let's finish this step strong and move on to the next part of our math journey!

Step 2 Solve for x

We've simplified our equation to:

9x = 18

Now, solving for 'x' is super straightforward. To isolate 'x', we need to get rid of the 9 that's multiplying it. How do we do that? We divide both sides of the equation by 9. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. This is a golden rule in algebra! It's like a scale – if you remove weight from one side, you need to remove the same amount from the other to keep it even. So, let's perform the division:

9x / 9 = 18 / 9

On the left side, 9x divided by 9 just leaves us with 'x'. On the right side, 18 divided by 9 is 2. So, we have:

x = 2

Fantastic! We've found the value of 'x'. It's like discovering a key piece of a puzzle. With 'x' in hand, we're one step closer to unlocking the final answer. But we're not done yet – we still need to find 'y'. So, let's carry this value of 'x' forward and use it to solve for 'y' in the next step. We're making great progress, guys!

Step 3 Substitute x to Find y

Now that we know x = 2, we can substitute this value into either of our original equations to solve for 'y'. It doesn't matter which equation we choose, but let's pick the first one because it looks a bit simpler:

2x + 5y = 19

We're going to replace 'x' with 2. This is like swapping out a piece in a Lego set – we're taking out 'x' and putting in its actual value. So, let's make the substitution:

2(2) + 5y = 19

Now we've got an equation with just 'y' as the unknown. This is much easier to handle! Let's simplify the equation. First, we multiply 2 by 2, which gives us 4. So, the equation becomes:

4 + 5y = 19

Our next goal is to isolate 'y'. We'll start by subtracting 4 from both sides of the equation. This will get rid of the 4 on the left side, bringing us closer to solving for 'y'. Remember, balance is key – we do the same thing to both sides. So, let's do the subtraction and see what we get!

Isolating y

Let's subtract 4 from both sides of the equation:

4 + 5y - 4 = 19 - 4

This simplifies to:

5y = 15

We're almost there! Now, to get 'y' by itself, we need to divide both sides of the equation by 5. This is the final step in isolating 'y'. We're using the same principle as before – whatever we do to one side, we do to the other. So, let's divide and see what the value of 'y' is:

5y / 5 = 15 / 5

This gives us:

y = 3

Awesome! We've found the value of 'y'. It's like finding the matching key to our second lock. Now we know both x = 2 and y = 3. We've conquered the main hurdle – solving the system of equations. But we're not quite at the finish line yet. Remember, the original question asked us to find the value of 5x + 7y. So, we need to take these values of 'x' and 'y' and plug them into that expression. Let's head to the final step and wrap this up!

Step 4 Calculate 5x + 7y

We've found that x = 2 and y = 3. Now, we need to find the value of the expression 5x + 7y. This is the final piece of the puzzle! To do this, we'll substitute the values of 'x' and 'y' into the expression. This is like fitting the last pieces into a jigsaw puzzle to reveal the complete picture. So, let's make the substitutions:

5x + 7y = 5(2) + 7(3)

Now, we just need to do the arithmetic. First, we'll perform the multiplications: 5 times 2 is 10, and 7 times 3 is 21. So, the expression becomes:

10 + 21

Finally, we add 10 and 21 together. This is the last calculation we need to do, and it will give us our final answer. So, let's add them up!

The Final Calculation

Adding 10 and 21, we get:

10 + 21 = 31

So, the value of 5x + 7y is 31. We did it! We've successfully navigated through the system of equations and found the answer. It's like reaching the summit of a challenging climb – the view from the top is always worth the effort! We started with two equations, eliminated a variable, solved for 'x', then solved for 'y', and finally, we calculated the value of the expression. That's quite a journey! But more importantly, you now have a solid understanding of how to tackle similar problems. Remember, math is all about practice, so keep honing your skills, and you'll become a system-of-equations master in no time!

Answer

The value of 5x + 7y is 31. So the correct answer is d. 31.

Conclusion

And there you have it, guys! We've successfully solved this system of equations problem step by step. Remember, the key to mastering these problems is to break them down into manageable steps. First, we eliminated a variable, then we solved for the remaining variable, and finally, we used those values to find the answer we were looking for. It's like building a house – you lay the foundation, put up the walls, and then add the roof. Each step is crucial to the final result. Math might seem daunting at times, but with a clear strategy and a bit of practice, you can conquer any challenge. So keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!