Ordered Pairs That Satisfy F(x) = (1/2)x - 15: Find The Set!

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Hey guys! Today, we're diving into a super important concept in mathematics: identifying ordered pairs that satisfy a given equation. Specifically, we're tackling the equation f(x) = (1/2)x - 15. This type of problem is fundamental to understanding functions and their graphical representations. So, let's break it down step by step and make sure we all get it.

Understanding the Basics of Ordered Pairs and Equations

Before we jump into solving the problem, let's quickly refresh some key concepts. An ordered pair is a set of two numbers written in the form (x, y), where 'x' represents the input value and 'y' represents the output value. In the context of a function, 'x' is the independent variable (often the input), and 'y' is the dependent variable (the output we get after applying the function to 'x').

The equation f(x) = (1/2)x - 15 defines a linear function. This means that for any given value of 'x', we can calculate the corresponding value of f(x), which is our 'y'. To check if an ordered pair satisfies this equation, we simply substitute the 'x' value into the equation and see if we get the 'y' value of the ordered pair as the output.

What Does it Mean to "Satisfy" an Equation?

When we say an ordered pair "satisfies" an equation, we mean that when we plug the x-value of the ordered pair into the equation and perform the calculation, the result we get is equal to the y-value of the ordered pair. In simpler terms, the equation holds true for that particular pair of x and y values. This concept is crucial for understanding how points relate to lines and curves on a graph. Each point on the graph of a function represents an ordered pair that satisfies the function's equation. Therefore, finding ordered pairs that satisfy an equation is like pinpointing specific locations on the graph of that equation. It helps us visualize the function's behavior and understand its properties.

How to Test an Ordered Pair

Testing an ordered pair is a straightforward process that involves plugging the x-value into the equation and checking if the resulting y-value matches the y-value in the ordered pair. This method is fundamental for verifying whether a point lies on the graph of a function or whether a set of values is a solution to an equation. By mastering this skill, you gain a deeper understanding of the relationship between algebraic expressions and their graphical representations. This ability is not only essential for solving mathematical problems but also for visualizing and interpreting data in various fields such as science, engineering, and economics. For example, in physics, you might use this method to check if a set of data points fits a particular physical law. In economics, you could verify whether certain economic indicators align with a predicted model.

Analyzing the Given Options

Now, let's take a look at the options provided and see which set of ordered pairs satisfies our equation, f(x) = (1/2)x - 15. We have four options, each containing five ordered pairs. To find the correct answer, we need to test each ordered pair in each set.

Option A: (-2, 16), (0, 15), (2, -14), (4, -13), (6, -12)

Option B: (-2, -16), (0, -15), (2, -14), (4, -13), (6, -12)

Option C: (-2, 16), (0, 15), (2, 14), (4, 13), (6, 12)

We'll go through each option systematically, plugging in the 'x' values into the equation and checking if the resulting 'y' values match the ones given in the ordered pairs. This methodical approach ensures that we don't miss any potential solutions and helps us understand how each 'x' value affects the output of the function. It's like conducting a mini-experiment for each ordered pair, where we're testing the equation's behavior under different conditions. This practice not only reinforces our understanding of the equation but also hones our analytical skills, preparing us to tackle more complex problems in the future.

Step-by-Step Solution: Testing Each Ordered Pair

Let's start with Option A and test each ordered pair one by one:

  1. (-2, 16):
    • f(-2) = (1/2)(-2) - 15 = -1 - 15 = -16. This does not match the y-value of 16. So, this ordered pair does not satisfy the equation.
  2. (0, 15):
    • f(0) = (1/2)(0) - 15 = 0 - 15 = -15. This does not match the y-value of 15. So, this ordered pair does not satisfy the equation.
  3. (2, -14):
    • f(2) = (1/2)(2) - 15 = 1 - 15 = -14. This matches the y-value of -14. So, this ordered pair satisfies the equation.
  4. (4, -13):
    • f(4) = (1/2)(4) - 15 = 2 - 15 = -13. This matches the y-value of -13. So, this ordered pair satisfies the equation.
  5. (6, -12):
    • f(6) = (1/2)(6) - 15 = 3 - 15 = -12. This matches the y-value of -12. So, this ordered pair satisfies the equation.

Even though some ordered pairs in Option A satisfy the equation, not all of them do. For a set to be the correct answer, all ordered pairs must satisfy the equation. This underscores the importance of thoroughness in mathematical problem-solving. We can't just stop after finding a few correct pairs; we need to verify each and every one to ensure that the entire set aligns with the given condition. This meticulous approach is a hallmark of mathematical rigor and is crucial for building confidence in our solutions. Moreover, it teaches us the value of attention to detail, a skill that is highly transferable to various aspects of life beyond mathematics.

Now, let's move on to Option B and repeat the process:

  1. (-2, -16):
    • f(-2) = (1/2)(-2) - 15 = -1 - 15 = -16. This matches the y-value of -16. So, this ordered pair satisfies the equation.
  2. (0, -15):
    • f(0) = (1/2)(0) - 15 = 0 - 15 = -15. This matches the y-value of -15. So, this ordered pair satisfies the equation.
  3. (2, -14):
    • f(2) = (1/2)(2) - 15 = 1 - 15 = -14. This matches the y-value of -14. So, this ordered pair satisfies the equation.
  4. (4, -13):
    • f(4) = (1/2)(4) - 15 = 2 - 15 = -13. This matches the y-value of -13. So, this ordered pair satisfies the equation.
  5. (6, -12):
    • f(6) = (1/2)(6) - 15 = 3 - 15 = -12. This matches the y-value of -12. So, this ordered pair satisfies the equation.

Woohoo! All the ordered pairs in Option B satisfy the equation f(x) = (1/2)x - 15. This means we've found our answer! But just to be extra sure, let's quickly look at Option C to see why it's not the correct answer.

Option C: (-2, 16), (0, 15), (2, 14), (4, 13), (6, 12)

We already tested (-2,16) and (0,15) in option A and we know that they don't work, so option C is incorrect.

The Answer: Option B

Therefore, the set of ordered pairs that satisfies the equation f(x) = (1/2)x - 15 is:

B. (-2, -16), (0, -15), (2, -14), (4, -13), (6, -12)

Key Takeaways

  • Ordered pairs represent input and output values of a function.
  • To check if an ordered pair satisfies an equation, substitute the 'x' value into the equation and see if the result matches the 'y' value.
  • A set of ordered pairs satisfies an equation only if all pairs in the set satisfy the equation.

I hope this explanation was helpful! Remember, practice makes perfect. The more you work with ordered pairs and equations, the easier it will become. Keep up the great work, guys! You got this! This skill is like a fundamental building block in mathematics, paving the way for more advanced concepts. By mastering this, you're not just solving equations; you're developing a way of thinking that's crucial for problem-solving in various fields. Whether you're designing a bridge, predicting stock market trends, or analyzing scientific data, the ability to relate inputs and outputs through mathematical equations is a powerful tool. So, keep practicing, keep exploring, and keep building your mathematical foundation! You'll be amazed at how far it can take you. Remember, the journey of learning mathematics is like climbing a mountain. Each concept you master is a step closer to the summit, offering a broader and more breathtaking view of the mathematical landscape. So, embrace the challenges, celebrate the small victories, and enjoy the climb!