Ordered Pairs On Line Y=2x+87: How To Find Them?

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Hey guys! Let's dive into a common math problem: figuring out which ordered pairs fit perfectly on the line described by the equation y = 2x + 87. It might sound a bit intimidating at first, but trust me, it's totally manageable. We're going to break it down step by step, making sure you understand not just the how, but also the why behind it. So, let’s get started and make math a little less mysterious and a lot more fun!

Understanding Ordered Pairs and Linear Equations

Before we jump into the nitty-gritty, let’s make sure we're all on the same page with the basics. Ordered pairs are simply pairs of numbers, usually written in the form (x, y), that represent a specific point on a coordinate plane. Think of it like a map reference: the first number (x) tells you how far to go horizontally, and the second number (y) tells you how far to go vertically. Now, a linear equation, like our y = 2x + 87, is an equation that, when graphed, forms a straight line. The cool thing is that every single point on that line can be represented by an ordered pair (x, y) that satisfies the equation.

In our case, the equation y = 2x + 87 is in what we call slope-intercept form. This form, y = mx + b, is super handy because it tells us two important things right away: the slope (m) and the y-intercept (b). The slope tells us how steep the line is (how much y changes for every change in x), and the y-intercept is the point where the line crosses the vertical y-axis. For our equation, the slope (m) is 2, and the y-intercept (b) is 87. This means the line gets steeper as x increases, and it crosses the y-axis way up at 87! Knowing these basics is crucial because it helps us visualize what we're working with. We're not just dealing with abstract numbers and symbols; we're dealing with a line that has a specific direction and position on a graph. This visual understanding will make it much easier to determine which ordered pairs belong on this line.

The Table: Additional Toppings vs. Pizza Cost

The question often comes with a table that relates the number of additional toppings on a pizza (n) to the total cost (c). This table is our treasure map! It gives us specific (n, c) pairs, and our mission is to figure out which of these pairs fit our linear equation. Remember, for an ordered pair to lie on the line, it needs to make the equation true. This means if we plug the x-value (in our case, 'n', the number of toppings) into the equation and do the math, we should get the y-value (in our case, 'c', the cost) that's in the ordered pair. If we do, we've found a winner! It's like fitting puzzle pieces together: the ordered pair has to fit the equation perfectly. If it doesn't, it's not on the line. Understanding this connection between the table and the equation is key to solving the problem. We're not just randomly guessing; we're using the equation as a test to see which points belong.

How to Determine Which Ordered Pairs Lie on the Line

Okay, let’s get down to the nitty-gritty of how to actually figure out which ordered pairs lie on the line y = 2x + 87. It's a pretty straightforward process, and once you've done it a couple of times, you'll be a pro! The main idea is to take each ordered pair from the table and plug the x and y values into the equation. If the equation holds true, then that ordered pair lies on the line. If it doesn't, then it's not part of our line's crew.

Step-by-Step Process

  1. Identify the Ordered Pairs: First, we need to clearly identify the ordered pairs (x, y) from the table. Remember, in our case, the number of additional toppings (n) is our x-value, and the total cost of the pizza (c) is our y-value. So, if the table shows 1 topping costing $89, our ordered pair is (1, 89).
  2. Substitute the Values: Now comes the fun part! Take the x and y values from your ordered pair and plug them into the equation y = 2x + 87. For our example (1, 89), we would substitute x with 1 and y with 89, giving us: 89 = 2(1) + 87
  3. Simplify the Equation: Next, we need to simplify the equation and see if both sides are equal. Let's continue with our example:
    • 2(1) = 2
    • 2 + 87 = 89 So, our equation simplifies to: 89 = 89
  4. Check for Equality: This is the moment of truth! If the two sides of the equation are equal, like in our example where 89 equals 89, then the ordered pair lies on the line. If the two sides are not equal, then the ordered pair does not lie on the line. Imagine it like this: the equation is a lock, and the ordered pair is the key. If the key fits (the equation is true), then the ordered pair is on the line. If the key doesn't fit (the equation is false), then it's not.
  5. Repeat for All Ordered Pairs: You'll need to repeat these steps for every single ordered pair in the table. It might seem a bit tedious, but it’s the only way to be sure which pairs belong on the line. Each pair is a separate puzzle to solve, and you're figuring out if it fits the line's equation.

Example Time!

Let's walk through a couple of examples to really solidify this process. Suppose our table gives us the following ordered pairs:

  • (0, 87)
  • (2, 91)
  • (3, 93)

Let's test each one using our equation, y = 2x + 87.

  • For (0, 87):
    • Substitute: 87 = 2(0) + 87
    • Simplify: 87 = 0 + 87
    • Check: 87 = 87 (True!) So, (0, 87) lies on the line.
  • For (2, 91):
    • Substitute: 91 = 2(2) + 87
    • Simplify: 91 = 4 + 87
    • Check: 91 = 91 (True!) So, (2, 91) lies on the line.
  • For (3, 93):
    • Substitute: 93 = 2(3) + 87
    • Simplify: 93 = 6 + 87
    • Check: 93 = 93 (True!) So, (3, 93) lies on the line.

In this example, all three ordered pairs lie on the line y = 2x + 87. But remember, that won't always be the case! Sometimes, you'll find pairs that don't fit, and that's perfectly normal. The key is to go through the process methodically for each pair to be sure.

Common Mistakes to Avoid

Now that we've covered the process, let's chat about some common hiccups that students often encounter. Knowing these pitfalls can help you steer clear of them and ace those math problems!

1. Incorrect Substitution

One of the most frequent mistakes is substituting the x and y values incorrectly. Remember, the ordered pair is always (x, y), so make sure you're plugging the x-value in for x in the equation and the y-value in for y. A simple way to avoid this is to clearly label your values before you start substituting. Write a little 'x' above the first number in the ordered pair and a 'y' above the second number. This visual cue can help prevent those mix-ups.

2. Arithmetic Errors

Another common trap is making arithmetic errors when simplifying the equation. This usually happens during the multiplication or addition steps. A small mistake here can throw off your entire calculation and lead to the wrong answer. The solution? Take your time and double-check your work! If you're doing the calculations by hand, write out each step clearly. If you're using a calculator, make sure you're entering the numbers correctly. It might seem tedious, but a few extra seconds of checking can save you from a lot of frustration.

3. Forgetting Order of Operations

Order of operations (PEMDAS/BODMAS) is crucial! You need to make sure you're performing operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). If you mix up the order, you'll end up with the wrong result. So, always remember your PEMDAS/BODMAS! A handy mnemonic like