Equivalent Systems Of Equations A Comprehensive Guide

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Hey guys! Today, we're diving deep into the fascinating world of solving systems of equations. It might sound intimidating, but trust me, it's like solving a puzzle, and who doesn't love a good puzzle? Specifically, we're going to tackle the question: Which system of equations is equivalent to the following system?

{5x2+6y2=507x2+2y2=10\left\{ \begin{array}{l} 5 x^2+6 y^2=50 \\ 7 x^2+2 y^2=10 \end{array} \right.

And we'll be looking at a potential equivalent system:

{5x2+6y2=50−21x2−6y2=10\left\{ \begin{array}{r} 5 x^2+6 y^2=50 \\ -21 x^2-6 y^2=10 \end{array} \right.

So, grab your thinking caps, and let's get started!

Understanding Equivalent Systems of Equations

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what "equivalent systems" actually means. In simple terms, equivalent systems are sets of equations that have the exact same solutions. Think of it like this: they might look different on the surface, but when you solve them, you'll end up with the same values for x and y.

There are a few key operations we can perform on a system of equations that will result in an equivalent system. These operations are like the magic tricks of algebra, allowing us to manipulate equations without changing their fundamental solutions. Here’s a breakdown of the most common ones:

  • Multiplying an equation by a non-zero constant: If you multiply both sides of an equation by the same number (other than zero), you're essentially just scaling the equation. This doesn't change the relationship between the variables, so the solutions remain the same. For example, if you have the equation x + y = 5, multiplying both sides by 2 gives you 2x + 2y = 10, which has the same solutions.

  • Adding a multiple of one equation to another: This is a super useful technique. If you take one equation and add a multiple of another equation to it, you're creating a new equation that is a combination of the original two. This new equation, when combined with one of the original equations, forms an equivalent system. For instance, if you have the system:

    \left\{ \begin{array}{l}

x + y = 5 \ x - y = 1 \end{array} \right. $

You can add the second equation to the first to get 2x = 6. The system:

$
\left\{
\begin{array}{l}

2x = 6 \ x - y = 1 \end{array} \right. $

is equivalent to the original.

  • Swapping the order of equations: This one's pretty straightforward. The order in which you write the equations in a system doesn't affect the solutions. So, if you have a system:

    \left\{ \begin{array}{l}

x + y = 5 \ x - y = 1 \end{array} \right. $

It's equivalent to:

$
\left\{
\begin{array}{l}

x - y = 1 \ x + y = 5 \end{array} \right. $

Understanding these operations is crucial because they're the tools we'll use to determine if two systems are equivalent. We'll be looking for ways to transform one system into the other using these techniques.

Analyzing the Given Systems

Okay, let's get back to our specific problem. We have two systems of equations. The original system is:

{5x2+6y2=507x2+2y2=10\left\{ \begin{array}{l} 5 x^2+6 y^2=50 \\ 7 x^2+2 y^2=10 \end{array} \right.

And the system we're comparing it to is:

{5x2+6y2=50−21x2−6y2=10\left\{ \begin{array}{r} 5 x^2+6 y^2=50 \\ -21 x^2-6 y^2=10 \end{array} \right.

The first thing we notice is that the first equation in both systems is exactly the same: 5x² + 6y² = 50. This is a good sign because it means we only need to focus on the second equation to see if the systems are equivalent. The key question now becomes: can we manipulate the second equation of the original system (7x² + 2y² = 10) to obtain the second equation of the comparison system (-21x² - 6y² = 10) using our allowed operations?

Let's break down the second equations and see what needs to happen. We need to transform 7x² + 2y² = 10 into -21x² - 6y² = 10. Looking at the x² terms, we see that 7 would need to be multiplied by -3 to get -21. Similarly, the y² term, 2, would need to be multiplied by -3 to get -6. This suggests that multiplying the entire equation 7x² + 2y² = 10 by -3 might be a good starting point.

If we multiply the entire equation 7x² + 2y² = 10 by -3, we get:

-3(7x² + 2y²) = -3(10)

Which simplifies to:

-21x² - 6y² = -30

Now, let’s compare this result, -21x² - 6y² = -30, with the second equation in the comparison system, which is -21x² - 6y² = 10. We can see that the left-hand sides are identical (-21x² - 6y²), but the right-hand sides are different (-30 vs. 10). This is a critical observation.

Determining Equivalence

So, what does this difference in the right-hand sides tell us? It tells us that the two systems are not equivalent. Why? Because we've shown that by applying a valid operation (multiplying an equation by a constant) to the original system, we can obtain a new equation whose left-hand side matches the second equation in the comparison system, but whose right-hand side does not.

In other words, if the systems were equivalent, multiplying the second equation in the original system by -3 should have resulted in the second equation of the comparison system. However, it resulted in a different constant term. This discrepancy means that the solutions to the original system and the comparison system will not be the same.

To drive this point home, let's think about what it means for a system of equations to have a solution. A solution is a pair of values (x, y) that satisfies both equations simultaneously. If the right-hand sides of the equations are different after applying a valid transformation, it means that any pair (x, y) that satisfies one system will not satisfy the other.

For example, imagine you find a pair (x, y) that satisfies 5x² + 6y² = 50 and 7x² + 2y² = 10. When you plug those values into the equation -21x² - 6y² = -30 (which we derived from the original system), the equation will hold true. However, when you plug those same values into -21x² - 6y² = 10 (the second equation of the comparison system), the equation will not hold true because the right-hand sides are different.

Therefore, we can confidently conclude that the comparison system is not equivalent to the original system.

Why This Matters: The Importance of Equivalent Systems

You might be wondering, “Okay, so the systems aren’t equivalent. Why is this such a big deal?” Well, the concept of equivalent systems is fundamental to solving more complex problems in algebra and beyond. Understanding how to manipulate equations while preserving their solutions is crucial for a variety of reasons:

  • Simplifying Systems: Often, we encounter systems of equations that are messy or difficult to solve in their original form. By using the operations that create equivalent systems, we can transform these complex systems into simpler, more manageable ones. This might involve eliminating variables, rearranging terms, or scaling equations to make coefficients easier to work with.
  • Solving for Variables: When solving for unknown variables, we often use techniques like substitution or elimination. These techniques rely on the principle of equivalent systems. We manipulate the equations to isolate variables, knowing that the solutions we find will also be the solutions to the original system.
  • Graphical Interpretation: Equivalent systems have the same graphical representation. If you were to graph the equations in both systems, you would see that they intersect at the same points. This visual representation reinforces the idea that equivalent systems have the same solutions.
  • Applications in Real-World Problems: Systems of equations are used to model a wide range of real-world situations, from mixing chemicals in a lab to optimizing financial investments. Being able to work with equivalent systems allows us to solve these problems efficiently and accurately.

In essence, the ability to recognize and create equivalent systems is a powerful tool in the mathematician's toolkit. It allows us to approach problems from different angles, simplify complexity, and ultimately find solutions.

Key Takeaways and Further Exploration

Let's recap the main points we've covered today:

  • Equivalent systems are sets of equations that have the same solutions.
  • We can create equivalent systems by multiplying an equation by a non-zero constant, adding a multiple of one equation to another, or swapping the order of equations.
  • To determine if two systems are equivalent, we can try to transform one system into the other using these operations.
  • If we apply a valid operation and the resulting system has different solutions, the original systems are not equivalent.

This concept of equivalent systems is a cornerstone of algebra, and mastering it will open doors to more advanced mathematical concepts. If you're eager to explore further, here are a few avenues you might consider:

  • Practice Solving Systems: The best way to solidify your understanding is to work through examples. Find systems of equations online or in textbooks and practice solving them using various techniques.
  • Explore Different Methods: Learn about methods like substitution, elimination, and matrix methods for solving systems of equations. Understanding these methods will give you a deeper appreciation for the power of equivalent systems.
  • Investigate Linear Algebra: Linear algebra is a branch of mathematics that deals extensively with systems of equations and their properties. Delving into linear algebra will provide a more formal and abstract understanding of equivalent systems.
  • Real-World Applications: Look for examples of how systems of equations are used in real-world applications. This will help you see the practical relevance of the concepts you're learning.

Keep practicing, keep exploring, and keep those mathematical gears turning! Understanding equivalent systems is a valuable skill that will serve you well in your mathematical journey. Until next time, happy problem-solving!

In conclusion, by carefully analyzing the given systems and applying the principles of equivalent equations, we determined that the comparison system is not equivalent to the original system. Remember, guys, understanding these fundamental concepts is key to tackling more complex mathematical challenges. Keep practicing, and you'll become a master problem-solver in no time!