Hey guys! Today, we're diving into the exciting world of solving equations with ordered pairs. It might sound intimidating, but trust me, it's like a fun puzzle! We're given a list of ordered pairs and a set of equations, and our mission is to find which pairs fit perfectly into each equation. Think of it as finding the right key for the right lock. So, let's get started and unlock the secrets of ordered pairs and equations!
Understanding Ordered Pairs and Equations
Before we jump into solving, let's make sure we're all on the same page about what ordered pairs and equations actually are. This is super important because it forms the foundation for everything else we'll be doing. Ordered pairs, those little guys in parentheses like (1, 2) or (-1, 2), are essentially coordinates on a graph. The first number, usually labeled as 'x', tells us how far to move horizontally, and the second number, 'y', tells us how far to move vertically. Think of it like giving directions on a map!
Now, equations are like mathematical sentences. They state that two expressions are equal. In our case, these equations will have two variables, like 'p' and 't' or 'x' and 'y'. This means we need two values to make the equation true – hence the ordered pairs! Our mission is to substitute the 'x' and 'y' values from the ordered pair into the equation and see if it balances out. If it does, we've found a solution! If not, we move on to the next pair. Remember, the goal here is to find the specific ordered pair that, when plugged into the equation, makes the left side equal to the right side. This process of substituting and checking is the key to solving these types of problems. So, let’s keep this understanding in mind as we move forward and tackle each equation one by one. Ready to become equation-solving pros? Let's do this!
CC1: Solving 4p - 3t = -2
Okay, let's tackle our first equation: 4p - 3t = -2. Remember, we have a list of ordered pairs: (-1, 2), (1, -2), (-2, 1), and (1, 2). Our mission is to figure out which one of these pairs, when plugged into the equation, makes it true. Here's the thing, in each ordered pair, the first number represents the value of 'p' and the second number represents the value of 't'. So, we're going to substitute these values into the equation and see what happens. Think of it like a little math experiment!
Let's start with the first ordered pair, (-1, 2). We'll replace 'p' with -1 and 't' with 2 in the equation. This gives us: 4*(-1) - 3*(2) = -2. Now, let's simplify the left side of the equation. 4 multiplied by -1 is -4, and -3 multiplied by 2 is -6. So, we have -4 - 6. What does that equal? It's -10. Now, let's compare this to the right side of our original equation, which is -2. Is -10 equal to -2? Nope! That means (-1, 2) is not a solution to this equation. But don't worry, that's just part of the process. We learn by trying!
Next, let's try the ordered pair (1, -2). We substitute 'p' with 1 and 't' with -2. This gives us: 4*(1) - 3*(-2) = -2. Let's simplify again. 4 multiplied by 1 is 4, and -3 multiplied by -2 is +6 (remember, a negative times a negative is a positive!). So, we have 4 + 6, which equals 10. Is 10 equal to -2? Definitely not! So, (1, -2) is also not a solution. We're learning what doesn't work, which is just as important as finding what does!
Now, let's move on to (-2, 1). We replace 'p' with -2 and 't' with 1. This gives us: 4*(-2) - 3*(1) = -2. Simplifying, 4 multiplied by -2 is -8, and -3 multiplied by 1 is -3. So, we have -8 - 3, which equals -11. Is -11 equal to -2? Nope, not this time either. But we're getting closer to understanding how these equations work!
Finally, let's try the ordered pair (1, 2). We substitute 'p' with 1 and 't' with 2. This gives us: 4*(1) - 3*(2) = -2. Simplifying, 4 multiplied by 1 is 4, and -3 multiplied by 2 is -6. So, we have 4 - 6. What does that equal? It's -2! And guess what? That's exactly what's on the right side of our original equation. We've found a match! So, the ordered pair (1, 2) is the solution to the equation 4p - 3t = -2. High five! See, it's like a puzzle, and we just found the right piece. Now, let's move on to the next equation and keep our problem-solving brains working!
CC2: Finding the Solution for 4x + y = -2
Alright, team! Let's move on to our next equation: 4x + y = -2. We're going to use the same strategy as before – plugging in our ordered pairs and seeing if they make the equation true. Remember, our options are (-1, 2), (1, -2), (-2, 1), and (1, 2). The first number in each pair is 'x', and the second number is 'y'. So, let's get to substituting and solving!
First up, let's try (-1, 2). We replace 'x' with -1 and 'y' with 2 in the equation. This gives us: 4*(-1) + 2 = -2. Okay, let's simplify. 4 multiplied by -1 is -4. So, we have -4 + 2. What does that equal? It's -2. And guess what? That's exactly what's on the right side of the equation! We've got a solution! The ordered pair (-1, 2) works perfectly for this equation. Awesome job! This shows how important it is to be systematic in our approach. By carefully substituting and simplifying, we can crack these equations.
Now, just for practice and to be thorough, let's quickly check the other ordered pairs as well. This is a good habit to get into because sometimes there might be more than one solution, or we might catch a mistake we made earlier.
Let's try (1, -2). Substituting, we get: 4*(1) + (-2) = -2. Simplifying, 4 multiplied by 1 is 4. So, we have 4 - 2, which equals 2. Is 2 equal to -2? Nope! So, (1, -2) is not a solution.
Next, let's check (-2, 1). Substituting, we get: 4*(-2) + 1 = -2. Simplifying, 4 multiplied by -2 is -8. So, we have -8 + 1, which equals -7. Is -7 equal to -2? Definitely not!
Finally, let's try (1, 2). Substituting, we get: 4*(1) + 2 = -2. Simplifying, 4 multiplied by 1 is 4. So, we have 4 + 2, which equals 6. Is 6 equal to -2? Nope, not a solution.
So, we've confirmed that only (-1, 2) is the solution for the equation 4x + y = -2. See how checking the other pairs helped us be extra sure of our answer? This is a great strategy to use in your own problem-solving. Now that we've conquered this equation, let's move on to the next one. We're building our equation-solving skills with every step!
CC3: Solving the Equation 5d - 4c = 13
Alright, let's keep the momentum going and dive into our next equation: 5d - 4c = 13. We're sticking with the same game plan: taking each ordered pair from our list – (-1, 2), (1, -2), (-2, 1), and (1, 2) – and plugging them into the equation to see if they fit. Remember, the first number in each pair is 'd', and the second number is 'c'. So, let's get those substitutions rolling!
Let's start with (-1, 2). We replace 'd' with -1 and 'c' with 2 in the equation. This gives us: 5*(-1) - 4*(2) = 13. Now, let's simplify. 5 multiplied by -1 is -5, and -4 multiplied by 2 is -8. So, we have -5 - 8. What does that equal? It's -13. Now, let's compare this to the right side of our original equation, which is 13. Is -13 equal to 13? Nope, not even close! So, (-1, 2) is not a solution for this equation. But that's okay, we're just warming up!
Next, let's try the ordered pair (1, -2). We substitute 'd' with 1 and 'c' with -2. This gives us: 5*(1) - 4*(-2) = 13. Let's simplify again. 5 multiplied by 1 is 5, and -4 multiplied by -2 is +8 (remember, a negative times a negative is a positive!). So, we have 5 + 8, which equals 13. And guess what? That's exactly what's on the right side of our original equation! We've found a solution! The ordered pair (1, -2) is a perfect fit for this equation. High five! It's so satisfying when the numbers line up just right.
But just like before, let's check the remaining ordered pairs to make sure we haven't missed anything and to get in some extra practice. This is a great way to build confidence in our problem-solving skills.
Let's move on to (-2, 1). We replace 'd' with -2 and 'c' with 1. This gives us: 5*(-2) - 4*(1) = 13. Simplifying, 5 multiplied by -2 is -10, and -4 multiplied by 1 is -4. So, we have -10 - 4, which equals -14. Is -14 equal to 13? Nope, not a match.
Finally, let's try the ordered pair (1, 2). We substitute 'd' with 1 and 'c' with 2. This gives us: 5*(1) - 4*(2) = 13. Simplifying, 5 multiplied by 1 is 5, and -4 multiplied by 2 is -8. So, we have 5 - 8. What does that equal? It's -3. Is -3 equal to 13? Nope, not this time either.
So, we've confirmed that the only solution for the equation 5d - 4c = 13 from our list of ordered pairs is (1, -2). We're really getting the hang of this! Each equation is like a new challenge, and we're rising to the occasion. Now, let's tackle our final equation and bring our A-game!
CC4: Determining the Solution for a + 4b = -7
Okay, guys, we've reached the final equation in our set: a + 4b = -7. We're going to use the same trusty method that's been working so well for us: plugging in the ordered pairs and checking if they make the equation true. Our ordered pairs are still (-1, 2), (1, -2), (-2, 1), and (1, 2). Remember, the first number in each pair represents 'a', and the second number represents 'b'. Let's do this!
Let's start with the ordered pair (-1, 2). We substitute 'a' with -1 and 'b' with 2. This gives us: -1 + 4*(2) = -7. Now, let's simplify. 4 multiplied by 2 is 8. So, we have -1 + 8. What does that equal? It's 7. Is 7 equal to -7? Nope, not even close. So, (-1, 2) is not the solution for this equation. But that's perfectly fine. We're learning with every attempt!
Now, let's try the ordered pair (1, -2). We substitute 'a' with 1 and 'b' with -2. This gives us: 1 + 4*(-2) = -7. Let's simplify. 4 multiplied by -2 is -8. So, we have 1 - 8. What does that equal? It's -7! And guess what? That's exactly what we have on the right side of our equation. We've found a solution! The ordered pair (1, -2) works for this equation. Awesome job! It's like we're becoming equation-solving detectives, finding the right clues to crack the case.
But, as always, let's make absolutely sure by checking the other ordered pairs. It's a fantastic way to reinforce our understanding and catch any potential mistakes.
Let's check (-2, 1). Substituting, we get: -2 + 4*(1) = -7. Simplifying, 4 multiplied by 1 is 4. So, we have -2 + 4, which equals 2. Is 2 equal to -7? Definitely not!
Finally, let's try the ordered pair (1, 2). Substituting, we get: 1 + 4*(2) = -7. Simplifying, 4 multiplied by 2 is 8. So, we have 1 + 8, which equals 9. Is 9 equal to -7? Nope, it's not a match.
So, we've confirmed that the only solution for the equation a + 4b = -7 from our list of ordered pairs is (1, -2). We did it! We've successfully navigated through all four equations and found the correct ordered pairs for each one. Give yourselves a pat on the back – you've earned it!
Conclusion: Mastering Ordered Pairs and Equations
Wow, guys! We've really been through it, haven't we? We've explored the world of ordered pairs and equations, learning how to find the perfect matches between them. We started by understanding what ordered pairs and equations actually mean, and then we jumped into the nitty-gritty of substituting values and simplifying expressions. We tackled four different equations, each with its own unique challenge, and we emerged victorious!
We discovered that solving equations with ordered pairs is like a puzzle – it requires careful attention to detail, a systematic approach, and a willingness to try different possibilities. We learned the importance of substituting values correctly, simplifying expressions accurately, and comparing the results to find the right solutions. We also saw how checking our answers and verifying our solutions can give us extra confidence in our work.
Remember, the key to mastering these skills is practice, practice, practice! The more you work with ordered pairs and equations, the more comfortable and confident you'll become. So, don't be afraid to take on new challenges, explore different types of equations, and keep honing your problem-solving abilities. You've got this! Keep up the fantastic work, and I can't wait to see what mathematical feats you'll accomplish next!