Does (3, -1) Satisfy 2x - 5y = -11? A Mathematical Exploration

Hey guys! Let's dive into a math problem that might seem tricky at first, but we'll break it down together step by step. Our main question is: Does the ordered pair (3, -1) satisfy the equation 2x - 5y = -11? We've got some options to consider: A) Yes, the ordered pair satisfies the equation. B) No, the ordered pair does not satisfy the equation. C) It's not possible to determine with just one ordered pair. And D) Yes.

This question falls under the realm of coordinate geometry and linear equations, which are fundamental concepts in mathematics. Understanding how to work with ordered pairs and equations is super important, not just for math class, but also for real-world applications like mapping, computer graphics, and even economics. So, let's get started and figure out the right answer!

Understanding Ordered Pairs and Equations

Before we jump into solving the problem, let's make sure we're all on the same page about what ordered pairs and equations actually mean. This is the cornerstone of solving this problem, so let's get cozy with these concepts.

An ordered pair, like (3, -1), is simply a set of two numbers written in a specific order. The first number, in this case 3, represents the x-coordinate, and the second number, -1, represents the y-coordinate. Think of it as a specific location on a graph, where the x-coordinate tells you how far to move horizontally, and the y-coordinate tells you how far to move vertically. These coordinates are the key to unlocking solutions in many mathematical problems.

Now, an equation is a mathematical statement that shows the equality between two expressions. In our case, the equation is 2x - 5y = -11. This equation describes a straight line on a graph. Any ordered pair that satisfies this equation will lie on that line. To satisfy the equation means that when you plug in the x and y values from the ordered pair into the equation, the left side of the equation will equal the right side. It’s like finding the perfect puzzle piece that fits just right!

In essence, we are dealing with a basic concept in algebra where we want to test if a given point lies on a given line. This is a quintessential problem in linear algebra and is foundational for more advanced topics. The beauty of mathematics is how these basic building blocks form complex structures, and this simple problem is a great illustration of that.

The Process of Verification: Substituting and Solving

Now that we understand the basic concepts, let's get to the heart of the problem. Our mission is to determine if the ordered pair (3, -1) satisfies the equation 2x - 5y = -11. To do this, we need to use a process called substitution. Think of it as replacing the generic 'x' and 'y' in our equation with the specific numbers from our ordered pair.

Here's how it works:

1. Identify the x and y values in the ordered pair. In (3, -1), x = 3 and y = -1. It’s crucial to keep the order correct, as switching them would change the entire meaning.
2. Substitute these values into the equation 2x - 5y = -11. This means we replace 'x' with 3 and 'y' with -1. So, our equation becomes 2(3) - 5(-1) = -11.
3. Simplify the left side of the equation using the order of operations (PEMDAS/BODMAS). First, we perform the multiplications: 2(3) = 6 and -5(-1) = 5. Now our equation looks like this: 6 + 5 = -11.
4. Evaluate the left side: 6 + 5 = 11. So now we have 11 = -11.
5. Compare the result with the right side of the equation. Does 11 equal -11? No, it doesn't! This is where we make our critical determination.

The act of substitution is a cornerstone of algebra. It's like translating a general rule (the equation) into a specific instance (the ordered pair). The simplicity of this process belies its power. It's a method that scales from basic algebra to complex calculus problems. The magic of this technique is in its universality.

Analyzing the Result: Does the Ordered Pair Satisfy the Equation?

We've gone through the substitution process, plugged in our values, and simplified the equation. We arrived at the statement 11 = -11. Now, what does this tell us? This is where our analytical skills come into play.

Clearly, 11 does not equal -11. These are two distinct numbers, one positive and one negative. This inequality is the key to answering our original question. Since the left side of the equation does not equal the right side after substituting the values from the ordered pair, we can conclude that the ordered pair (3, -1) does not satisfy the equation 2x - 5y = -11.

Think of it like trying to fit a square peg into a round hole. The ordered pair, in this case, is the square peg, and the equation is the round hole. They just don't match! The numbers don't align, and the equation remains unbalanced. The beauty of mathematics is in this precise matching, where everything must fit perfectly.

This result also has a visual interpretation. Remember that the equation 2x - 5y = -11 represents a straight line on a graph. The fact that the ordered pair (3, -1) does not satisfy the equation means that the point represented by this ordered pair does not lie on that line. It's somewhere else on the coordinate plane, not a part of the line defined by our equation. This geometric connection adds another layer to our understanding.

Why Other Options Are Incorrect

Now that we've determined the correct answer, let's quickly look at why the other options are incorrect. This is a crucial step in problem-solving, as it solidifies our understanding and helps us avoid similar mistakes in the future.

• Option A) Sim, o par ordenado satisfaz a equação. (Yes, the ordered pair satisfies the equation.) We've already shown that this is incorrect. Our substitution and simplification clearly demonstrated that the ordered pair does not satisfy the equation.
• Option C) Não é possível determinar apenas com um par ordenado. (It is not possible to determine with just one ordered pair.) This is also incorrect. We were able to definitively determine whether the ordered pair satisfies the equation by substituting the values and checking for equality. One ordered pair is sufficient to test if it lies on a specific line.
• Option D) Sim. (Yes.) This option is too vague and doesn't provide a complete answer. While it might seem like it aligns with option A, it lacks the crucial context of stating that the ordered pair satisfies the equation. In a mathematical context, precision is key.

By eliminating these incorrect options, we reinforce our understanding of the problem and the solution process. It's like building a strong foundation for future mathematical explorations.

Conclusion: The Correct Answer and Its Significance

So, after our detailed exploration, we've arrived at the definitive answer: B) No, the ordered pair does not satisfy the equation. We reached this conclusion by understanding the concepts of ordered pairs and equations, performing the substitution process, simplifying the equation, and analyzing the result.

This exercise might seem simple, but it highlights a fundamental skill in mathematics: the ability to verify solutions. It's not enough to just find an answer; we need to be able to prove that our answer is correct. This verification process builds confidence and ensures accuracy in our mathematical endeavors. The importance of this cannot be overstated.

Furthermore, this problem illustrates the connection between algebra and geometry. The equation 2x - 5y = -11 represents a line, and the ordered pair (3, -1) represents a point. Determining whether the ordered pair satisfies the equation is equivalent to determining whether the point lies on the line. This connection between algebraic equations and geometric shapes is a powerful concept that underpins much of mathematics. It’s a seamless blend of concepts.

Understanding how to work with ordered pairs and equations is essential for a wide range of mathematical applications. From graphing lines and solving systems of equations to more advanced topics like calculus and linear algebra, these fundamental concepts are the building blocks for more complex ideas. So, mastering this skill is a significant step in your mathematical journey.

So, there you have it, guys! We've successfully tackled this problem and learned some valuable mathematical lessons along the way. Keep practicing, keep exploring, and keep the mathematical fire burning!