Does (3, -1) Satisfy 2x - 5y = -11 A Step-by-Step Analysis

Hey guys! Today, we're diving into a fun math problem that involves ordered pairs and equations. We're going to figure out if the ordered pair (3, −1) satisfies the equation 2x − 5y = −11. And to spice things up, we'll also explore what happens when we substitute x = 1 into the equation. So, grab your calculators (or your mental math muscles!) and let's get started!

Understanding Ordered Pairs and Equations

Before we jump into the problem, let's quickly recap what ordered pairs and equations are all about. An ordered pair is simply a pair of numbers written in a specific order, usually represented as (x, y). The first number, x, represents the point's position on the horizontal axis (the x-axis), and the second number, y, represents the point's position on the vertical axis (the y-axis). Think of it as a coordinate on a map! An equation, on the other hand, is a mathematical statement that shows the equality between two expressions. It often involves variables (like x and y) and constants (numbers). In our case, the equation is 2x − 5y = −11, which is a linear equation, meaning its graph would be a straight line.

Linear equations are fundamental in mathematics and have vast applications in various fields, from physics and engineering to economics and computer science. They describe relationships where the change in one variable is proportional to the change in another. The standard form of a linear equation is typically written as Ax + By = C, where A, B, and C are constants, and x and y are variables. In our equation, 2x − 5y = −11, A is 2, B is -5, and C is -11. Understanding this form helps us easily identify the coefficients and constants, which are crucial for solving the equation and analyzing its properties. For instance, the coefficients A and B are related to the slope and intercepts of the line represented by the equation. Mastering linear equations is not just about solving for x and y; it's about grasping the relationships they represent and how these relationships can model real-world scenarios. By substituting values into the equation, we can check if specific points lie on the line, which is exactly what we'll be doing with the ordered pair (3, -1). This process of substitution and verification is a cornerstone of algebraic problem-solving and is essential for understanding the solutions of equations and systems of equations.

Verifying the Ordered Pair (3, −1)

Now, let's tackle the main question: Does the ordered pair (3, −1) satisfy the equation 2x − 5y = −11? To find out, we need to substitute the values of x and y from the ordered pair into the equation and see if it holds true. Remember, in the ordered pair (3, −1), x = 3 and y = −1. So, let's plug these values into the equation:

2x − 5y = −11 2(3) − 5(−1) = −11 6 + 5 = −11 11 = −11

Oops! It looks like we hit a snag. The equation doesn't hold true because 11 is definitely not equal to −11. This means the ordered pair (3, −1) does not satisfy the equation 2x − 5y = −11. So, the correct answer to our initial question is B) Não, o par ordenado não satisfaz a equação (No, the ordered pair does not satisfy the equation).

Substitution is a fundamental technique in algebra, allowing us to replace variables with specific values to evaluate expressions and solve equations. In this case, substituting x = 3 and y = -1 into the equation 2x - 5y = -11 is a straightforward application of this technique. However, the real power of substitution lies in its broader applications. For example, it’s used extensively in solving systems of equations, where we might solve one equation for one variable and then substitute that expression into another equation to find the value of the remaining variable. Similarly, substitution plays a key role in calculus, where it’s used in integration and differentiation to simplify complex expressions. Understanding the nuances of substitution, such as ensuring the correct signs and order of operations, is crucial for accurate problem-solving. In our example, the mistake could easily have been made by not properly accounting for the negative sign in -5y when y = -1. Attention to detail and a solid grasp of arithmetic are essential when applying substitution in algebraic manipulations. Furthermore, substitution is not just a mathematical tool; it's a way of thinking. It allows us to break down complex problems into smaller, manageable parts and to explore the relationships between variables and constants. By mastering substitution, we gain a powerful tool for tackling a wide range of mathematical challenges.

Exploring the Equation with x = 1

Now, let's switch gears and see what happens when we substitute x = 1 into the equation 2x − 5y = −11. This will give us a different perspective on the equation and help us understand how changing one variable affects the other. So, let's replace x with 1:

2x − 5y = −11 2(1) − 5y = −11 2 − 5y = −11

Now, we have a new equation with only one variable, y. To solve for y, we need to isolate it on one side of the equation. Let's start by subtracting 2 from both sides:

2 − 5y − 2 = −11 − 2 −5y = −13

Next, we'll divide both sides by −5 to get y by itself:

−5y / −5 = −13 / −5 y = 13/5

So, when x = 1, y = 13/5. This means the ordered pair (1, 13/5) is a solution to the equation 2x − 5y = −11. This exercise demonstrates how we can find different solutions to an equation by choosing a value for one variable and solving for the other.

Solving equations for a specific variable involves a series of algebraic manipulations aimed at isolating that variable on one side of the equation. The key principle behind these manipulations is maintaining the balance of the equation; whatever operation is performed on one side must also be performed on the other. In our example, after substituting x = 1, we had the equation 2 − 5y = −11. To isolate y, we first subtracted 2 from both sides, which is a valid operation because it doesn't disrupt the equality. Then, we divided both sides by -5. Each of these steps is guided by the goal of simplifying the equation and bringing y closer to being on its own. It's crucial to follow the order of operations (PEMDAS/BODMAS) and to pay close attention to signs, especially when dealing with negative numbers. Errors in these details can easily lead to incorrect solutions. Furthermore, the process of solving equations reinforces the concept of inverse operations. Subtraction is the inverse of addition, and division is the inverse of multiplication. By applying these inverse operations in the correct order, we can systematically unravel the equation and find the value of the unknown variable. This methodical approach to solving equations is a fundamental skill in algebra and is essential for tackling more complex mathematical problems. It also provides a foundation for understanding more advanced concepts like functions and graphing.

Analyzing the Alternatives

Now that we've explored the problem thoroughly, let's revisit the alternatives and see which ones make sense in light of our findings:

A) Sim, o par ordenado satisfaz a equação. (Yes, the ordered pair satisfies the equation.) - Incorrect. We showed that (3, −1) does not satisfy the equation. B) Não, o par ordenado não satisfaz a equação. (No, the ordered pair does not satisfy the equation.) - Correct. This aligns with our calculations. C) Com x = 1, a equação não... (With x = 1, the equation does not...) - This option is incomplete, but we know that with x = 1, we can solve for y and find a solution (1, 13/5).

So, the correct alternative is B. We've not only answered the question but also gained a deeper understanding of how ordered pairs and equations work together. Great job, guys!

Conclusion

In conclusion, we've successfully determined that the ordered pair (3, −1) does not satisfy the equation 2x − 5y = −11. We did this by substituting the values of x and y into the equation and verifying that the equation did not hold true. Additionally, we explored what happens when we substitute x = 1 into the equation, finding that it leads to a new solution (1, 13/5). This exercise highlights the importance of understanding how to work with ordered pairs, equations, and substitution in algebra. These are fundamental concepts that form the building blocks for more advanced mathematical topics. By practicing these skills, we can become more confident and proficient in solving a wide range of mathematical problems. Keep up the great work, everyone, and remember that math can be fun when we approach it with curiosity and a willingness to explore!

This problem also underscores the importance of precision and attention to detail in mathematics. Even a small error, like misinterpreting a negative sign, can lead to an incorrect answer. It’s crucial to double-check our work and to be methodical in our approach. Furthermore, this exercise demonstrates the interconnectedness of mathematical concepts. Our understanding of ordered pairs, equations, and substitution techniques all came together to solve this problem. This interconnectedness is a hallmark of mathematics, and as we progress in our studies, we’ll see how different concepts build upon each other to form a cohesive and powerful system of knowledge. By mastering these fundamental skills and cultivating a mindset of precision and attention to detail, we can unlock the beauty and utility of mathematics and apply it to a wide range of real-world problems.

So, the next time you encounter an ordered pair and an equation, remember the steps we took today. Substitute, simplify, and solve! And most importantly, don't be afraid to explore and have fun with math!