Is (5,-7) A Solution? Check This System Of Equations!

Dec 3, 2025 by Admin 54 views

Hey there, math explorers! Ever stared at a bunch of equations and an ordered pair like (5,-7) and wondered, "Is this specific pair actually a solution to this whole system?" Well, you're in the right place, because today we're going to dive deep into how to figure out exactly that! We're talking about a super important concept in algebra: determining if an ordered pair like our (5,-7) truly satisfies every single equation in a system of equations. It might sound a bit complex at first glance, but trust me, it's actually a pretty straightforward and logical process once you get the hang of it. We'll be using our specific example, the ordered pair (5,-7) and the system of equations given as x+y=-2 and 5x+3y=18, to show you exactly how it's done. So, grab your thinking caps, get ready to do some substitution, and let's unravel this mathematical mystery together! By the end of this article, you'll be a pro at checking ordered pairs against systems of equations, making sure you never miss a step.

What's the Big Deal with Systems of Equations, Anyway?

So, what's the big deal with systems of equations, anyway? Basically, guys, a system of equations is just a fancy term for two or more equations that are all hanging out together, and we're looking for values that work for all of them simultaneously. Think of it like a mathematical puzzle where each equation is a unique clue, and the solution is the one set of values (in our case, an ordered pair like (x, y)) that makes every single clue true. Why do we even bother with these? Well, systems of equations are incredibly powerful tools for solving real-world problems. Imagine you're a business owner trying to figure out how many different types of products to stock based on their cost and your budget, or perhaps you're a scientist needing to balance chemical reactions with multiple variables. These aren't just abstract numbers; they represent tangible situations and help us model complex relationships. Understanding them is fundamental to fields ranging from economics to engineering.

An ordered pair, like the (5,-7) we're focusing on today, is simply a pair of numbers where the order matters. The first number (5 in our specific case) always represents the x-coordinate, and the second number (-7) represents the y-coordinate. When we say an ordered pair is a solution to a system of equations, we mean that if you substitute the x-value and the y-value from that pair into every single equation in the system, each equation will result in a true statement. For example, if you plug x=5 and y=-7 into an equation and it simplifies to 2 = 2, that's a true statement! If, however, it simplifies to 2 = 5, that's a false statement. If even one equation comes up false, then our ordered pair is not a solution to the entire system. It's an all-or-nothing game, folks! This strict requirement is what makes systems of equations so precise and useful. Understanding systems of equations and how to identify their solutions is fundamental not just for passing your math class, but for developing critical thinking skills that apply to countless scenarios. From basic budgeting to complex engineering challenges, the ability to analyze multiple constraints (equations) and find a common point that satisfies them all (the ordered pair solution) is absolutely invaluable. We're not just doing math here; we're building problem-solving muscles! So, before we jump into the nitty-gritty of testing (5,-7) with x+y=-2 and 5x+3y=18, it's super important to grasp this core concept: a solution to a system must satisfy every single equation. No exceptions! This strong foundation will make the rest of our journey much smoother and help you confidently check any ordered pair for any system of equations you encounter.

The Game Plan: How to Check an Ordered Pair

Alright, guys, now that we've got a handle on what systems of equations are and why solutions matter, let's talk about the game plan for how to check an ordered pair to see if it's truly a solution. It’s a pretty straightforward process, almost like a step-by-step checklist, and it involves a simple but powerful mathematical operation: substitution. When you're faced with an ordered pair, like our trusty (5,-7), and a system of equations, such as x+y=-2 and 5x+3y=18, your main goal is to methodically substitute the x and y values from the ordered pair into each equation one by one. This isn't a race; it's a careful verification process that demands attention to detail. Every single equation in the system must be individually tested and confirmed.

Here's the step-by-step breakdown of how we're going to check an ordered pair effectively:

Identify your ordered pair and system: First things first, clearly identify the (x, y) values you're testing (in our specific case, x=5 and y=-7) and the system of equations you're working with (x+y=-2 and 5x+3y=18). Being organized from the start prevents silly mistakes and ensures you're using the correct numbers throughout your calculations. Write them down clearly if it helps.
Test with the first equation: Take the x and y values from your ordered pair and substitute them into the first equation in your system. Perform the arithmetic carefully, following the order of operations. After the substitution and calculation, you'll end up with a statement. This statement will either be true (e.g., 5 = 5) or false (e.g., 5 = 10). If this first equation results in a false statement, you can stop immediately! The ordered pair is not a solution to the system. No need to proceed further.
Test with the second equation (and any others!): If the first equation resulted in a true statement, you must move on to the next equation in the system. Just because it works for one doesn't mean it works for all! Again, substitute the x and y values from your ordered pair into this second equation and calculate the result. Just like before, you'll see if it's true or false. If there are more equations in the system, repeat this step for every single one of them. Each must independently be verified as true.
Make your final verdict: This is where you decide. If every single equation in the system results in a true statement after you substitute the ordered pair's values, then — congratulations! — your ordered pair IS a solution to the system of equations. However, if even one single equation (whether it's the first, second, or tenth!) results in a false statement, then unfortunately, your ordered pair IS NOT a solution to the system. It's that simple, but also that strict. The key here is consistency across all equations. Don't stop at the first true statement unless that was the only equation; you must verify every equation. This systematic approach ensures you don't miss anything and provides a clear, undeniable path to determine if an ordered pair like (5,-7) is indeed the answer we're looking for in our specific system of equations. This method is foolproof if followed precisely.

Step 1: Plug and Play with the First Equation

Alright, math enthusiasts, let's roll up our sleeves and tackle the first equation in our system with our given ordered pair, (5,-7). Remember, our first equation is x+y=-2. The beauty of this step lies in its straightforwardness: we're simply going to plug in the x and y values from (5,-7) into this equation. So, for (5,-7), our x value is 5 and our y value is -7. We're literally replacing the x and y letters with their corresponding numbers. This process of substitution is a foundational skill in algebra, allowing us to evaluate expressions and verify statements with specific numerical inputs. It's like giving our algebraic expression a direct command, asking it, "What happens when x is 5 and y is -7?"

Here’s how it breaks down for the first equation, x+y=-2:

Original First Equation: x + y = -2
Substitute x=5 and y=-7: (5) + (-7) = -2
Perform the addition: When you add a positive number and a negative number, you essentially subtract the smaller absolute value from the larger one and keep the sign of the larger number. So, 5 - 7 becomes... -2.
Simplify and Check: -2 = -2

Now, take a good, hard look at that final statement: -2 = -2. Is that a true statement? Absolutely, it is! -2 definitely equals -2. This is fantastic news because it means our ordered pair (5,-7) satisfies the first equation, x+y=-2. This is a crucial initial success! If it hadn't, say we got 5 = -2 after the substitution, then we could immediately stop right there and declare (5,-7) not a solution to the system. But since it works for the first equation, we must continue to the second. This step is crucial for two reasons: firstly, it’s the initial filter. If the ordered pair doesn’t work for the first equation, there’s no need to even bother with the rest, saving us precious time and effort. Secondly, it establishes confidence. When you see a true statement like -2 = -2, you know you're on the right track for that specific equation. It’s like clearing the first hurdle in a race; you’re still running, but you’ve passed the initial test. We want to be absolutely certain that (5,-7) is a solution to the system, and that means it has to be a team player, working perfectly with every single equation. So far, so good for our ordered pair (5,-7) and our first equation x+y=-2. One down, one to go, guys! Keep that focus sharp as we move on to the next challenge, ensuring our potential solution stands up to all scrutiny.

Step 2: Don't Stop There! Test the Second Equation

Alright, math explorers, remember our game plan? We just successfully verified that our ordered pair (5,-7) works perfectly for the first equation, x+y=-2. But here’s the crucial part: we absolutely cannot stop there! A solution to a system of equations has to make every single equation in that system true. If it only works for one, it's like having one key that fits only one lock in a double-locked door—it’s not enough to get you all the way through to the treasure! So, with that in mind, let’s power through and test the second equation in our system, which is 5x+3y=18. We're going to apply the exact same substitution method we used before, plugging in x=5 and y=-7 from our ordered pair (5,-7). This consistent application of the ordered pair's values to each equation is what allows us to confidently verify the entire system.

Let’s break down the substitution for this second equation, 5x+3y=18:

Original Second Equation: 5x + 3y = 18
Substitute x=5 and y=-7: Now we replace x with 5 and y with -7. Remember to use parentheses for clarity, especially when dealing with multiplication and negative numbers: 5(5) + 3(-7) = 18
Perform the multiplications: According to the order of operations (PEMDAS/BODMAS), we do multiplication before addition. So, 5 * 5 gives us 25, and 3 * -7 gives us -21. The equation now looks like: 25 + (-21) = 18
Perform the addition: Adding 25 and -21 is the same as subtracting 21 from 25. This gives us 4.
Simplify and Check: 4 = 18

Now, let's stare hard at that final statement: 4 = 18. Is that true? Uh-oh, guys, it's a resounding false! Four definitely does not equal eighteen. This is where the rubber meets the road. Even though our ordered pair (5,-7) worked beautifully for the first equation (x+y=-2), it completely failed the test for the second equation (5x+3y=18). This single false statement means that (5,-7) is not a solution to the entire system of equations. It's a deal-breaker! You see, the ordered pair must be a loyal friend to all equations in the system. If it betrays even one, it's out! This step is where many students might mistakenly stop if they only test the first equation, but as you can see, testing all equations is absolutely fundamental. The moment you hit a false statement, you know your ordered pair is not the solution you're looking for. It's an unequivocal answer, allowing us to confidently move to our final verdict. This meticulousness is what makes you a master of checking ordered pairs in systems of equations. It demonstrates a thorough understanding of what a true solution entails, ensuring no stone is left unturned in our mathematical investigation. This critical verification step solidifies our understanding of systems of equations and how they operate.

The Verdict: Is (5,-7) a True Solution?

So, we’ve arrived at the moment of truth, guys! We've meticulously followed our game plan, plugging in our ordered pair (5,-7) into each equation in our system: x+y=-2 and 5x+3y=18. Let's quickly recap our findings to deliver the final verdict on whether (5,-7) is a true solution to this system of equations. Remember, for an ordered pair to be a true solution, it must satisfy both equations simultaneously. If it falls short on even one, it's a no-go for the entire system.

For the first equation, x+y=-2:

We substituted x=5 and y=-7 into the equation.
This gave us 5 + (-7) = -2, which quickly simplified to -2 = -2.
This was a TRUE statement! So far, so good. Our ordered pair (5,-7) successfully passed the test for the first equation, making it a potential candidate for a solution.

Now, for the second equation, 5x+3y=18:

We substituted x=5 and y=-7 into this equation.
This resulted in 5(5) + 3(-7) = 18, which became 25 + (-21) = 18, and finally simplified to 4 = 18.
This was a FALSE statement! Ouch! Our ordered pair (5,-7) did not satisfy the second equation, immediately invalidating its status as a solution to the system.

Based on these definitive results, we can confidently conclude that the ordered pair (5,-7) is NOT a solution to the given system of equations. Why not? Because, as we emphasized from the very beginning, for an ordered pair to be considered a true solution to a system of equations, it must make every single equation in that system true. It's like needing a specific key to open two separate locks on a treasure chest. If the key only opens one lock but not the other, you still can't get to the treasure, right? In our scenario, (5,-7) fit the first "lock" (x+y=-2), but it definitely didn't fit the second "lock" (5x+3y=18). Therefore, it fails the ultimate test for the entire system. This principle is absolutely fundamental to understanding systems of equations and their solutions. It’s not about partial credit; it’s about complete consistency. If you ever find yourself checking an ordered pair and one equation comes up false, you don't even need to continue checking any further equations if there were more – you already have your answer: not a solution. This clarity helps you efficiently work through more complex problems in the future, saving you time and preventing unnecessary calculations. You now possess the knowledge to swiftly determine if a given ordered pair holds up as a true solution across multiple conditions, a valuable skill in any mathematical context.

Phew! We've gone on quite the mathematical adventure today, haven't we? We started by asking a simple, yet profoundly important question: "Is (5,-7) a solution to the system of equations x+y=-2 and 5x+3y=18?" And through our careful, step-by-step process of substitution and verification, we've arrived at a definitive answer. Remember, the core takeaway here, guys, is that for an ordered pair to be a true solution, it needs to be a consistent truth-teller for all equations in the system. It's not enough to just make one of them happy; every single equation must be satisfied. This skill of checking ordered pairs against systems of equations is more than just a classroom exercise; it builds a critical foundation for solving much more intricate problems in algebra, geometry, and beyond. So, next time you see an ordered pair and a system, you'll know exactly how to approach it. Keep practicing, keep questioning, and keep mastering these awesome math concepts! You've got this, and you're well on your way to becoming a mathematical mastermind!