Ordered Pair (3, -1) In Equation 2x - 5y = -11: Solution Explained
Hey guys! Ever stumbled upon an equation and wondered if a specific set of numbers could actually make it true? That's exactly what we're diving into today. We're going to break down the concept of ordered pairs and how they relate to equations, using the question: Does the ordered pair (3, -1) satisfy the equation 2x - 5y = -11? It might sound intimidating at first, but trust me, it's a lot simpler than it looks. So, buckle up, and let's get started!
Understanding Ordered Pairs and Equations
Before we jump into solving the problem, let's quickly recap what ordered pairs and equations are all about. An ordered pair, like (3, -1), is simply a set of two numbers written in a specific order. The first number usually represents the x-coordinate, and the second number represents the y-coordinate on a graph. Think of it as a precise location on a map, where the order matters – (3, -1) is a different spot than (-1, 3).
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 represents a line on a graph, and any ordered pair that satisfies the equation will lie on that line. So, how do we check if an ordered pair satisfies an equation? That's where the magic happens – we substitute the x and y values from the ordered pair into the equation and see if it holds true. If both sides of the equation are equal after the substitution, then the ordered pair is a solution, meaning it satisfies the equation. If not, then it's not a solution. This method is a fundamental concept in algebra and is used extensively to solve various problems involving linear equations and systems of equations.
Imagine the equation as a puzzle, and the ordered pair as a potential piece. Our goal is to see if this piece fits perfectly into the puzzle. If it does, we've found a solution. If not, we need to look for another piece. This analogy helps to visualize the process of substituting and verifying solutions in equations. Understanding this basic principle is crucial for tackling more complex mathematical problems later on. So, let's keep this in mind as we move forward and apply it to our specific question.
The Substitution Method: Plugging in the Values
Alright, guys, let's get our hands dirty and put this ordered pair (3, -1) to the test! The core of our solution lies in the substitution method. This method is a powerful tool in algebra, allowing us to check if a given set of values satisfies an equation. In our case, we're going to substitute the x and y values from our ordered pair into the equation 2x - 5y = -11. Remember, the first number in the ordered pair is the x-value, and the second number is the y-value. So, in (3, -1), x = 3 and y = -1.
Now, here comes the fun part! We're going to replace the 'x' in the equation with 3 and the 'y' with -1. Our equation 2x - 5y = -11 will transform into 2(3) - 5(-1) = -11. Notice how we've carefully replaced the variables with their corresponding values. It's crucial to pay attention to the signs here, as a small mistake can throw off the entire calculation. Next, we need to simplify the left side of the equation by performing the multiplication operations. 2 multiplied by 3 gives us 6, and -5 multiplied by -1 gives us +5 (remember, a negative times a negative is a positive!). So, our equation now looks like 6 + 5 = -11.
We're almost there! The next step is to add the numbers on the left side of the equation. 6 + 5 equals 11. So, our equation now reads 11 = -11. This is the critical point where we need to compare both sides of the equation. Are they equal? Do the numbers match? If they do, the ordered pair satisfies the equation, and we have a solution! If they don't, the ordered pair is not a solution, and we need to move on. So, let's take a close look and see what we've got.
Evaluating the Result: Does the Equation Hold True?
Okay, guys, we've reached the moment of truth! After substituting and simplifying, we arrived at the statement 11 = -11. Now, let's take a good look. Does 11 equal -11? Absolutely not! These are two distinct numbers on the number line, one positive and one negative. This inequality tells us something important: 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 – it just doesn't work. Similarly, the values from the ordered pair (3, -1) don't fit into the relationship defined by the equation 2x - 5y = -11. When we substitute these values, the equation doesn't balance out; the left side doesn't equal the right side. This means that the point represented by the ordered pair (3, -1) does not lie on the line represented by the equation 2x - 5y = -11. In graphical terms, if we were to plot this point and draw the line, we would see that the point is not on the line.
This result is crucial because it reinforces the concept of solutions to equations. Not every ordered pair will be a solution to a given equation. The ordered pair must satisfy the equation, meaning it must make the equation true when substituted. In our case, (3, -1) fails this test. So, based on our evaluation, we can confidently conclude that the ordered pair (3, -1) is not a solution to the equation 2x - 5y = -11. This understanding is fundamental for solving more complex problems involving systems of equations and inequalities.
The Correct Answer and Why It Matters
So, after our step-by-step journey of substitution and evaluation, we've arrived at a clear conclusion: the ordered pair (3, -1) does not satisfy the equation 2x - 5y = -11. This means 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).
You might be wondering, why is this important? Well, understanding how to check if an ordered pair satisfies an equation is a fundamental skill in algebra and beyond. It's the building block for solving systems of equations, which are used to model real-world scenarios in various fields like science, engineering, economics, and computer science. For instance, imagine you're trying to find the point where two lines intersect on a graph. This point represents the solution to a system of two equations, and you would use the same principles we've discussed today to find it. Or, think about balancing chemical equations, where you need to find values that satisfy multiple equations simultaneously.
Furthermore, this process reinforces the importance of precision and attention to detail in mathematics. A small error in substitution or simplification can lead to a completely wrong answer. This exercise demonstrates the need to be methodical and double-check your work. Mastering these basic skills will set you up for success in more advanced math courses and in problem-solving situations in general. So, guys, keep practicing these concepts, and you'll be well on your way to becoming math whizzes!
Practice Makes Perfect: Further Exploration
Alright, guys, now that we've nailed this problem, let's talk about how to solidify your understanding and become even more confident with ordered pairs and equations. The key, as with most things in math, is practice, practice, practice! The more you work with these concepts, the more natural they'll become.
One great way to practice is to try similar problems. Look for equations and ordered pairs and test whether they satisfy the equations. You can find these in textbooks, online resources, or even create your own! Try changing the numbers in the equation or the ordered pair and see how it affects the outcome. For example, what if we had the equation 2x - 5y = 11? Would (3, -1) satisfy that equation? What if we changed the ordered pair to (3, 1)? How would that change the result? Experimenting with different values will give you a deeper understanding of the relationships between equations and ordered pairs.
Another helpful exercise is to visualize these concepts graphically. Remember, an equation like 2x - 5y = -11 represents a line on a graph, and an ordered pair represents a point. If an ordered pair satisfies the equation, it means the point lies on the line. Try graphing the equation and plotting the point (3, -1) to visually confirm that it doesn't lie on the line. This graphical representation can make the concept more intuitive and help you remember it better.
Finally, don't hesitate to seek out additional resources and support if you're struggling. There are tons of excellent websites, videos, and tutorials online that can explain these concepts in different ways. Talk to your teacher or classmates, form study groups, and help each other out. Remember, learning math is a journey, and it's okay to ask for help along the way. So, keep exploring, keep practicing, and keep challenging yourselves. You've got this!
By mastering the fundamentals of ordered pairs and equations, you'll not only ace your math exams but also gain a valuable skill set that will serve you well in various aspects of life. So, let's keep the momentum going and continue our mathematical adventures!
