Solving Y=-3x+6 And Y=9 System Of Equations

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Hey guys! Today, we're diving into the exciting world of solving systems of equations. Think of it like detective work, where we're trying to find the mystery values that make two equations true at the same time. We'll break down a specific example step-by-step, so you can become a system-solving pro. So, let's get started and make math a little less mysterious!

The Problem: Unlocking the Values of x and y

Our mission, should we choose to accept it, is to find the solution to the following system of equations:

y = -3x + 6
y = 9

What does this actually mean? Well, we have two equations, each with two variables, x and y. Our goal is to find the specific values for x and y that satisfy both equations simultaneously. In other words, we need to find the point where the lines represented by these equations intersect on a graph. This point of intersection is the solution to our system.

But how do we find this magical point? That's where the fun begins! There are a few methods we can use, but for this problem, the substitution method is going to be our best friend. It's a straightforward and elegant way to crack the code. We are going to deep dive into the Substitution Method, highlighting why it works so well in situations like this. Basically, since we already know what y equals in the second equation (y = 9), we can substitute that value directly into the first equation. This is like having a key piece of the puzzle that perfectly fits into another part of the puzzle, revealing more of the picture. We'll walk through each step meticulously, ensuring every detail is crystal clear. We’ll also explore common pitfalls to avoid and tricks for verifying your answers. This isn't just about finding the solution to this specific problem; it's about equipping you with a versatile tool you can use to solve countless other systems of equations. By mastering the substitution method, you'll gain a deeper understanding of how equations interact and how to manipulate them to uncover hidden solutions. So, stay tuned as we unravel the mysteries of substitution and transform you into a system-solving wizard!

The Substitution Solution: Cracking the Code

The substitution method is particularly handy when one of the equations is already solved for one variable (like our second equation, y = 9). Here's how it works:

  1. Substitute: Since we know y = 9, we can substitute 9 for y in the first equation:

    9 = -3x + 6

    This is the magic step! We've transformed the first equation from having two variables (x and y) into an equation with just one variable (x). Now we can solve for x!

  2. Solve for x: Let's isolate x in the equation. First, subtract 6 from both sides:

    9 - 6 = -3x + 6 - 6
3 = -3x

    Next, divide both sides by -3:

    3 / -3 = -3x / -3
-1 = x

    We've found our x value! x = -1.

  3. Find y: We already know y = 9 from the second equation. Awesome!

  4. The Solution: The solution to the system of equations is the ordered pair (x, y) = (-1, 9).

Isn't that neat? By substituting the known value of y from the second equation into the first, we successfully navigated our way to finding the value of x. This single move transformed a seemingly complex problem into a straightforward equation that we could easily solve. This is the power of substitution – it simplifies the problem by reducing the number of variables we need to deal with at one time. Now that we have the value of x, finding y was a piece of cake because the second equation already gave it to us! This highlights another advantage of the substitution method: when one equation is already solved for a variable, it provides a direct pathway to finding the other variable once you've solved for the first one. In the next section, we’ll take this solution and put it to the test. We’ll delve into the crucial step of verification, ensuring that our hard-earned answer is indeed the correct one. This step is essential in building confidence in your problem-solving skills and preventing those pesky errors that can sometimes creep in. So, let’s not stop here; let’s solidify our understanding by verifying our solution and ensuring we've truly cracked the code!

Verification: Double-Checking Our Work

It's always a good idea to verify our solution to make sure we didn't make any mistakes. To do this, we substitute our values for x and y back into both original equations.

  • Equation 1: y = -3x + 6

    Substitute x = -1 and y = 9:

    9 = -3(-1) + 6
9 = 3 + 6
9 = 9  // This is true!

  • Equation 2: y = 9

    Substitute y = 9:

    9 = 9  // This is also true!

Since our solution satisfies both equations, we can confidently say that it's correct!

This process of verification is not just a formality; it's a critical step in ensuring accuracy and solidifying your understanding of the problem. Think of it as the final piece of the puzzle that confirms everything fits together perfectly. By plugging our solution back into the original equations, we’re essentially performing a reverse check, making sure that the values we found truly satisfy the conditions set by the system. This practice is particularly important in mathematics because it helps catch any arithmetic errors or missteps that might have occurred during the solving process. It also reinforces the core concept of what a solution to a system of equations actually means – a set of values that makes all the equations in the system true simultaneously. Furthermore, the act of verifying cultivates a habit of thoroughness and attention to detail, qualities that are invaluable not only in mathematics but in all aspects of problem-solving. It teaches you to be your own critic, to question your results, and to seek confirmation before accepting an answer as final. So, always remember to take that extra step and verify your solutions. It’s a small investment of time that can save you from big headaches later on and transform you into a more confident and capable problem-solver.

The Answer: Choosing the Right Option

Looking back at the answer choices, we see that option C. (-1, 9) matches our solution.

So, the correct answer is C. (-1, 9). We did it!

This journey of solving the system of equations has not only led us to the correct answer but has also illuminated the power and elegance of the substitution method. We've witnessed how a strategic substitution can transform a complex-looking problem into a manageable equation, and we've learned the importance of verification in ensuring the accuracy of our solution. But the learning doesn't stop here! The world of systems of equations is vast and fascinating, and there are many more techniques and concepts to explore. Think about the graphical interpretation of these solutions – how each equation represents a line, and the solution is the point where they intersect. Imagine systems with three or more equations and variables, where the solutions represent points in higher dimensions. These concepts might seem daunting now, but with a solid foundation in methods like substitution, you'll be well-equipped to tackle them. So, keep practicing, keep exploring, and keep challenging yourself. Mathematics is a journey of discovery, and each problem you solve is a step forward on that journey. And who knows, maybe one day you'll be the one teaching others how to crack the codes of complex equations! For now, let’s move on to the next section where we recap the key steps and explore some additional tips for tackling similar problems.

Key Takeaways and Extra Tips

Let's recap the main steps we took to solve this system of equations:

  1. Identify the Best Method: In this case, substitution was ideal because one equation was already solved for y. But remember, guys, there are other methods like elimination, which might be a better fit for other problems.

  2. Substitute: Replace the variable in one equation with its equivalent expression from the other equation.

  3. Solve: Solve the resulting equation for the remaining variable.

  4. Back-Substitute: Plug the value you found back into one of the original equations to solve for the other variable.

  5. Verify: Check your solution in both original equations.

Here are a few extra tips to keep in mind:

  • Stay Organized: Write your steps clearly and neatly to avoid mistakes.
  • Double-Check Arithmetic: Simple errors can throw off your whole solution, so pay attention to detail.
  • Practice Makes Perfect: The more systems of equations you solve, the better you'll become at recognizing the best method and avoiding common pitfalls.

Solving systems of equations is a fundamental skill in mathematics with applications in various fields, from engineering to economics. The ability to find solutions to these systems allows us to model and understand real-world relationships, make predictions, and solve complex problems. Think about it: businesses use systems of equations to optimize production costs, engineers use them to design structures and circuits, and economists use them to forecast market trends. By mastering this skill, you're not just learning a mathematical concept; you're equipping yourself with a powerful tool for analyzing and solving problems in a wide range of contexts. The key is to approach each problem strategically, identify the underlying relationships, and apply the appropriate techniques to unravel the solutions. And remember, even if you encounter challenges along the way, don't get discouraged. Every mistake is an opportunity to learn and grow. Keep practicing, keep exploring, and keep pushing the boundaries of your understanding. The world of mathematics is full of exciting discoveries waiting to be made, and you have the potential to make significant contributions. So, embrace the challenge, sharpen your skills, and unlock the power of systems of equations to solve the problems of tomorrow.

Conclusion: You're a System-Solving Star!

Great job, everyone! You've successfully navigated the world of systems of equations and learned how to solve them using the substitution method. Remember, math isn't about memorizing formulas; it's about understanding concepts and developing problem-solving skills. Keep practicing, keep asking questions, and keep exploring the amazing world of mathematics!

So, guys, what's the next math challenge you're going to conquer? Keep up the awesome work!