Ordered Pair (3, -1): Does It Satisfy 2x - 5y = -11?

by Luna Greco 53 views

Hey guys! Let's dive into a fun math problem today. We're going to figure out if the ordered pair (3, -1) satisfies the equation 2x - 5y = -11. It might sound a bit intimidating, but trust me, it's easier than it looks! We'll break it down step by step and make sure everyone understands. So, grab your thinking caps, 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. This will give us a solid foundation and make the rest of the process super clear. Think of it as setting the stage for our mathematical performance!

What is an Ordered Pair?

An ordered pair is simply a set of two numbers written in a specific order, usually inside parentheses and separated by a comma. For example, (3, -1) is an ordered pair. The first number, in this case, 3, represents the x-coordinate, and the second number, -1, represents the y-coordinate. Ordered pairs are often used to represent points on a coordinate plane, which is a grid system used in graphs. The x-coordinate tells you how far to move horizontally from the origin (the point where the x and y axes intersect), and the y-coordinate tells you how far to move vertically. So, when we talk about the ordered pair (3, -1), we're essentially talking about a specific location on a graph.

What is an Equation?

An equation is a mathematical statement that shows the equality between two expressions. It typically involves variables (like x and y), constants (numbers), and mathematical operations (like addition, subtraction, multiplication, and division). The equation 2x - 5y = -11 is an example of a linear equation in two variables. It's called linear because if you were to graph all the ordered pairs (x, y) that satisfy this equation, you would get a straight line. In essence, an equation is a rule that defines a relationship between variables. Our goal is to see if the ordered pair (3, -1) fits into this rule.

Why are Ordered Pairs and Equations Important?

Ordered pairs and equations are fundamental concepts in mathematics and have a wide range of applications in real life. They help us model relationships between quantities, solve problems involving unknown values, and make predictions based on data. For instance, in physics, equations can describe the motion of objects, and ordered pairs can represent the position of an object at a specific time. In economics, equations can model supply and demand, and ordered pairs can represent the price and quantity of a product. Understanding these concepts opens the door to a deeper understanding of the world around us.

Now that we have a good grasp of what ordered pairs and equations are, we're ready to tackle the main problem. Let's move on to the next step: substituting the values from the ordered pair into the equation.

Substituting Values into the Equation

Alright, let's get our hands dirty with some actual calculations! The heart of this problem lies in substituting the values from our ordered pair (3, -1) into the equation 2x - 5y = -11. This process will help us determine if the ordered pair is a solution to the equation. Think of it like trying to fit a key into a lock – we're checking if the ordered pair "fits" the equation.

The Substitution Process Explained

Remember, in the ordered pair (3, -1), 3 is the x-coordinate, and -1 is the y-coordinate. The equation 2x - 5y = -11 has two variables, x and y. To substitute the values, we simply replace x with 3 and y with -1 in the equation. This is a crucial step, so let's take it slow and make sure we do it correctly. It's like following a recipe – accurate measurements are key to a delicious result!

So, after substitution, the equation will look like this: 2(3) - 5(-1) = -11. Notice how we've replaced x with 3 and y with -1. The parentheses are important here because they indicate multiplication. Now we have a numerical expression that we can simplify to see if it equals -11. It's like we've translated the algebraic language into a numerical puzzle that we can solve.

Step-by-Step Breakdown

Let's break down the calculation step by step to ensure clarity:

  1. Multiply 2 by 3: 2 * 3 = 6
  2. Multiply -5 by -1: -5 * -1 = 5. Remember, a negative number multiplied by a negative number gives a positive number. This is a fundamental rule of arithmetic that we need to keep in mind.
  3. Substitute these results back into the equation: 6 + 5 = -11

Now we have a simpler equation: 6 + 5 = -11. We're almost there! The next step is to perform the addition on the left side of the equation. It's like we're peeling away the layers of the problem to reveal the core truth.

Common Mistakes to Avoid

When substituting values, it's easy to make small errors that can lead to an incorrect answer. Here are a few common mistakes to watch out for:

  • Incorrectly substituting values: Make sure you substitute the x-coordinate for x and the y-coordinate for y. Mixing them up can change the whole outcome.
  • Sign errors: Pay close attention to negative signs, especially when multiplying. A misplaced negative sign can throw off the entire calculation.
  • Order of operations: Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This ensures you perform the calculations in the correct sequence.

By understanding the substitution process and being mindful of potential errors, we can confidently move forward and determine if the ordered pair satisfies the equation. Let's head to the next section where we'll simplify the equation and see what we get!

Simplifying the Equation

Okay, guys, we've substituted the values, and now it's time to simplify the equation. This is where we crunch the numbers and see if the left side of the equation matches the right side. Think of it as the moment of truth – we're about to find out if our key fits the lock!

Performing the Addition

We left off with the equation 6 + 5 = -11. The next step is to add 6 and 5. This is a straightforward addition, and it gives us 11. So now our equation looks like this: 11 = -11. It's like we're slowly revealing the answer, piece by piece.

Comparing Both Sides of the Equation

Now comes the crucial part: comparing the two sides of the equation. We have 11 on the left side and -11 on the right side. Are these two values equal? Absolutely not! 11 is a positive number, while -11 is a negative number. They are opposites on the number line. This is a key observation that will lead us to our final answer. It's like we've discovered a mismatch – the two sides of the equation are not in harmony.

Understanding Equality in Equations

In an equation, the two sides must be equal for the equation to be true. If the left side and the right side have different values, the equation is not satisfied. This is a fundamental concept in algebra. It's like a balance scale – for the scale to be balanced, the weights on both sides must be the same. In our case, the scale is tipped because 11 and -11 are not equal.

Common Mistakes in Simplification

Just like with substitution, there are a few common mistakes to watch out for when simplifying equations:

  • Arithmetic errors: Double-check your addition, subtraction, multiplication, and division. Simple arithmetic mistakes can lead to incorrect conclusions.
  • Misinterpreting signs: Pay close attention to positive and negative signs. A sign error can change the entire outcome.
  • Skipping steps: It's tempting to rush through the simplification process, but skipping steps can increase the likelihood of making a mistake. Take your time and write out each step clearly.

By carefully performing the addition and comparing the two sides of the equation, we've reached a critical point. We've discovered that 11 is not equal to -11. Now, let's move on to the final step: drawing our conclusion and selecting the correct answer.

Drawing the Conclusion

We've reached the final destination! We've substituted the values, simplified the equation, and compared both sides. Now, it's time to draw our conclusion. Based on our calculations, does the ordered pair (3, -1) satisfy the equation 2x - 5y = -11? Let's break it down.

Analyzing Our Results

We found that after substituting x = 3 and y = -1 into the equation, we ended up with 11 = -11. This statement is clearly false. 11 and -11 are not equal. This means that the ordered pair (3, -1) does not satisfy the equation 2x - 5y = -11. It's like we've tried to fit our key into the lock, and it just doesn't turn. The pieces don't fit together.

Connecting Back to the Question

The original question asked whether the ordered pair (3, -1) satisfies the equation 2x - 5y = -11. We've now definitively answered that question: no, it does not. The ordered pair does not lie on the line represented by the equation. It's like the point represented by the ordered pair is located somewhere off the line on a graph.

Evaluating the Answer Choices

Now, let's look at the answer choices provided and select the correct one:

A) Não é possível determinar apenas com um par ordenado. (It is not possible to determine with only one ordered pair.) B) Sim, o par ordenado satisfaz a equação. (Yes, the ordered pair satisfies the equation.) C) Não, o par ordenado não satisfaz a equação. (No, the ordered pair does not satisfy the equation.) D) Sim,

Based on our work, the correct answer is C) Não, o par ordenado não satisfaz a equação. We've shown through our calculations that the ordered pair (3, -1) does not make the equation 2x - 5y = -11 true. It's like we've found the perfect match for our answer based on the evidence we've gathered.

Final Thoughts

We've successfully solved the problem! We started with the question, broke it down into smaller steps, and arrived at the correct answer. This process highlights the power of substitution and simplification in solving algebraic problems. It's like we've followed a logical path, step by step, to reach our destination.

Remember, practice makes perfect! The more you work with ordered pairs and equations, the more comfortable you'll become with these concepts. So, keep practicing, keep exploring, and keep having fun with math!