Solve: Is (3,5) Or (7,6) A Solution To Y=2x-8?
Hey guys! Ever get that feeling when you're staring at an equation and a bunch of ordered pairs, and you're thinking, "Which one actually works here?" Well, you're definitely not alone! In this article, we're going to break down exactly how to figure out which ordered pair is a solution to the equation y = 2x - 8. We'll take a close look at what an ordered pair really means, how to plug it into the equation, and walk through the steps to find the correct answer. Get ready to become a pro at solving these types of problems! We’ll go through each option, (3, 5) and (7, 6), and see if they fit the equation like a perfect puzzle piece. So, grab your thinking caps, and let's dive in!
Understanding Ordered Pairs and Equations
Before we jump into solving, let's make sure we're all on the same page about the basics. An ordered pair, like (3, 5) or (7, 6), is simply a set of two numbers that are written in a specific order. The first number always represents the x-value, and the second number represents the y-value. Think of it like a map: the x-value tells you how far to go horizontally, and the y-value tells you how far to go vertically. These pairs are super important in math because they help us pinpoint exact locations on a graph or see if a point fits a particular relationship described by an equation.
Now, what about the equation y = 2x - 8? This equation is like a rule that connects x and y. It tells us that if we take any x-value, multiply it by 2, and then subtract 8, we'll get the corresponding y-value. So, when we're trying to figure out if an ordered pair is a solution to the equation, we're really asking: "Does this x-value and y-value follow the rule?" Does plugging these numbers into the equation make it true? If it does, then we've found a solution! This concept is the backbone of solving linear equations and understanding how different points relate to each other on a graph. We'll be using this idea throughout our problem-solving process, so make sure it clicks!
Testing the Ordered Pair (3, 5)
Alright, let's get our hands dirty and start testing! Our first ordered pair is (3, 5). Remember, the first number is the x-value, and the second number is the y-value. So, in this case, x = 3 and y = 5. Our equation is y = 2x - 8. To see if this ordered pair is a solution, we're going to plug these values into the equation and see if both sides are equal. This is like checking if the numbers fit the rule perfectly. If they do, we've found a winner! If not, we move on to the next contender.
So, let's substitute x = 3 and y = 5 into our equation:
5 = 2(3) - 8
Now, we need to simplify the right side of the equation. First, we multiply 2 by 3, which gives us 6:
5 = 6 - 8
Next, we subtract 8 from 6, which gives us -2:
5 = -2
Uh oh! This doesn't look right, does it? We've ended up with 5 equaling -2, which is definitely not true. This means that the ordered pair (3, 5) does not satisfy the equation y = 2x - 8. It's like trying to fit a square peg into a round hole – it just doesn't work. So, (3, 5) is not a solution. But don't worry, we've got another ordered pair to test. Let's see if (7, 6) fares any better!
Testing the Ordered Pair (7, 6)
Okay, time for round two! This time, we're testing the ordered pair (7, 6). Just like before, the first number is the x-value, and the second number is the y-value. So, here, x = 7 and y = 6. We're going to follow the same process as before: plug these values into the equation y = 2x - 8 and see if both sides are equal. This is the crucial step in determining if this ordered pair is a solution to our equation.
Let's substitute x = 7 and y = 6 into the equation:
6 = 2(7) - 8
Now, let's simplify the right side. First, we multiply 2 by 7, which gives us 14:
6 = 14 - 8
Next, we subtract 8 from 14, which gives us 6:
6 = 6
Bingo! This looks much better. We've ended up with 6 equaling 6, which is absolutely true. This means that the ordered pair (7, 6) does indeed satisfy the equation y = 2x - 8. It's like finding the missing piece of the puzzle – everything fits perfectly. So, (7, 6) is a solution to the equation. We've successfully found one of the solutions, and now we can confidently say which ordered pair works!
Conclusion: Identifying Solutions to Equations
So, let's wrap it all up! We started with the question: Which of the ordered pairs (3, 5) and (7, 6) is a solution to the equation y = 2x - 8? We broke down the process step by step, showing you exactly how to test each ordered pair and determine if it fits the equation. By plugging in the x and y values from each ordered pair into the equation and simplifying, we could see whether the equation held true.
We found that when we plugged in (3, 5), the equation didn't balance out. The left side and the right side were not equal, meaning (3, 5) is not a solution. However, when we tested (7, 6), the equation worked perfectly! Both sides were equal, confirming that (7, 6) is indeed a solution to the equation y = 2x - 8. This method of substituting values into an equation is a fundamental skill in algebra and helps us understand the relationships between variables.
Remember, this process is super useful for solving all sorts of equations. Whether you're dealing with linear equations, quadratic equations, or more complex functions, the basic principle remains the same: plug in the values, simplify, and see if the equation holds true. Keep practicing, and you'll become a master at identifying solutions to equations in no time! You've got this!
