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

Hey guys! Let's dive into the world of ordered pairs and linear equations. We've got a question on our hands: Does the ordered pair (3, -1) satisfy the equation 2x - 5y = -11? Sounds like a math adventure, right? So, let's break it down step by step.

Understanding Ordered Pairs and Equations

First, let’s clarify what we’re dealing with. An ordered pair, like (3, -1), is a set of two numbers where the order matters. The first number, in this case, 3, represents the x-coordinate, and the second number, -1, represents the y-coordinate. These coordinates tell us a specific location on a coordinate plane, which is essentially a grid formed by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical). The x-coordinate tells us how far to move horizontally from the origin (the point where the axes intersect), and the y-coordinate tells us how far to move vertically.

A linear equation, on the other hand, is an equation that can be written in the form Ax + By = C, where A, B, and C are constants, and x and y are variables. When graphed on a coordinate plane, a linear equation forms a straight line. This line represents all the possible ordered pairs (x, y) that satisfy the equation. In our case, the equation is 2x - 5y = -11. This means we're looking for ordered pairs that, when plugged into this equation, make the equation true. Every point on the line represented by this equation is a solution to the equation.

So, how do we determine if a specific ordered pair satisfies a given linear equation? The process is pretty straightforward. We substitute the x-coordinate of the ordered pair for x in the equation and the y-coordinate for y. Then, we simplify the equation and check if the left-hand side (LHS) equals the right-hand side (RHS). If they are equal, the ordered pair satisfies the equation; if they are not, it doesn’t. This is a fundamental concept in algebra and is crucial for solving systems of equations, graphing lines, and understanding the relationship between algebraic equations and their geometric representations. The beauty of this method lies in its simplicity and directness. It provides a concrete way to verify whether a given point lies on the line represented by the equation, or, in more general terms, whether a given set of values satisfies a given equation. This concept extends beyond linear equations and applies to various types of equations and functions in mathematics.

Plugging in the Values: A Step-by-Step Guide

Now, let’s get our hands dirty and actually plug in the values from our ordered pair (3, -1) into the equation 2x - 5y = -11. This is where the rubber meets the road, guys! We're going to take those abstract symbols and turn them into concrete numbers. Remember, the first number in the ordered pair (3, -1) is the x-coordinate, which is 3, and the second number is the y-coordinate, which is -1. So, we're going to replace every 'x' in our equation with a 3 and every 'y' with a -1.

Our equation is 2x - 5y = -11. Let’s substitute the values: 2(3) - 5(-1) = -11. See what we did there? We replaced the 'x' with a 3 and the 'y' with a -1. Now, it’s just a matter of simplifying the equation using the order of operations, which, if you recall, is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order is crucial for ensuring we arrive at the correct answer. We perform the operations in this sequence to maintain consistency and avoid ambiguity in mathematical calculations.

First, we perform the multiplications: 2 * 3 = 6 and -5 * -1 = 5. Remember, a negative times a negative gives a positive. This is a fundamental rule of arithmetic that we need to keep in mind. Our equation now looks like this: 6 + 5 = -11. Next, we perform the addition: 6 + 5 = 11. So, our equation simplifies to 11 = -11. This is the moment of truth! We've simplified the left-hand side of the equation as much as we can, and now we need to compare it to the right-hand side. Is 11 equal to -11? The answer is a resounding no! 11 and -11 are distinct numbers; they lie on opposite sides of zero on the number line and have different signs. This result tells us something very important: the ordered pair (3, -1) does not satisfy the equation 2x - 5y = -11. In other words, the point represented by the ordered pair (3, -1) does not lie on the line represented by the equation 2x - 5y = -11. This entire process highlights the power of substitution and simplification in algebra. By carefully replacing variables with their corresponding values and following the order of operations, we can determine whether a given set of values is a solution to an equation. This technique is a cornerstone of algebraic problem-solving and is used extensively in various mathematical contexts.

The Verdict: Does It Satisfy the Equation?

After substituting the values and simplifying, we arrived at the equation 11 = -11. Guys, this is clearly not true! 11 is not equal to -11. One is a positive number, and the other is a negative number. They are on opposite sides of zero on the number line. So, what does this mean for our original question? It means that the ordered pair (3, -1) does not satisfy the equation 2x - 5y = -11. It’s as simple as that. We've used a straightforward, step-by-step process to arrive at a definitive answer.

Think of it this way: if the ordered pair (3, -1) satisfied the equation, plugging in the values would have resulted in a true statement, like 5 = 5 or -2 = -2. But since we ended up with 11 = -11, which is false, we know that (3, -1) is not a solution. This process is like a mathematical detective story. We’re given a clue (the ordered pair) and a mystery (the equation), and we use our algebraic tools to solve the case. In this case, the verdict is clear: the ordered pair is not the culprit!

This concept is fundamental in algebra and has far-reaching applications. When we say an ordered pair satisfies an equation, we mean that the point represented by that ordered pair lies on the graph of the equation. In the case of a linear equation, the graph is a straight line. So, if (3, -1) satisfied 2x - 5y = -11, the point (3, -1) would be on the line that represents the equation. But since it doesn't, we know that the point is somewhere else on the coordinate plane, not on that particular line. Understanding this connection between ordered pairs, equations, and graphs is crucial for visualizing mathematical concepts and solving problems in various fields, from physics and engineering to economics and computer science. The ability to determine whether a point satisfies an equation is a foundational skill that opens doors to more advanced mathematical topics.

Choosing the Correct Option

So, looking back at our original question, we were asked to choose the correct alternative. We had three options: a) Yes, the ordered pair satisfies the equation; b) No, the ordered pair does not satisfy the equation; and c) Sim, but with x = 1 it is not possible to determine only with a pair. Based on our calculations and the logical reasoning we’ve walked through, we know that the correct answer is b) No, the ordered pair does not satisfy the equation. We've shown definitively that when we substitute the values from the ordered pair (3, -1) into the equation 2x - 5y = -11, we do not get a true statement. The left-hand side and the right-hand side of the equation are not equal, so the ordered pair is not a solution.

Option a) is incorrect because we’ve proven that the ordered pair does not satisfy the equation. Option c) is also incorrect because it introduces an extraneous condition (x = 1) that is not relevant to the question. Our focus was solely on the ordered pair (3, -1), and we were able to determine whether it satisfied the equation without any additional information or conditions. The beauty of mathematics lies in its precision and clarity. We can use logical steps and established rules to arrive at a correct answer with certainty. In this case, we followed the process of substitution and simplification, and the result clearly showed that the ordered pair (3, -1) is not a solution to the equation 2x - 5y = -11. This exercise underscores the importance of careful calculation and logical deduction in mathematical problem-solving. It also highlights the connection between algebraic concepts and their practical application in determining the validity of mathematical statements. The ability to accurately assess whether an ordered pair satisfies an equation is a fundamental skill that is essential for success in more advanced mathematics and related fields.

So, there you have it! We've successfully navigated the world of ordered pairs and equations and answered our question with confidence. Keep practicing, guys, and you'll become math whizzes in no time!