Solve Y=2x+5: Find The Correct Ordered Pair

Hey guys! Today, we're diving into a super common type of problem you'll see in algebra: figuring out which ordered pairs are solutions to a given equation. In this case, our equation is y = 2x + 5. An ordered pair is simply a set of two numbers, usually written as (x, y), that represent a point on a graph. A solution to an equation is an ordered pair that makes the equation true when you plug the x and y values into it. Let's break down how to tackle these problems and make sure you nail them every time!

Understanding Ordered Pairs and Solutions

Before we jump into the specifics of the equation y = 2x + 5, let's make sure we're all on the same page about what ordered pairs and solutions really mean. Think of an ordered pair (x, y) as a set of instructions. The first number, x, tells you how far to move horizontally on a graph (positive to the right, negative to the left). The second number, y, tells you how far to move vertically (positive up, negative down). So, the ordered pair (1, 7) means: start at the origin (the point (0, 0)), move 1 unit to the right, and then move 7 units up. That's the location of the point (1, 7) on the graph.

Now, what does it mean for an ordered pair to be a solution to an equation? It means that if you substitute the x-value and the y-value from the ordered pair into the equation, the equation will be true. Let's say we have the equation y = 2x + 5. If we plug in x = 1 and y = 7, we get 7 = 2(1) + 5, which simplifies to 7 = 7. This is a true statement, so the ordered pair (1, 7) is a solution to the equation. On the other hand, if we plug in x = 7 and y = 1, we get 1 = 2(7) + 5, which simplifies to 1 = 19. This is a false statement, so the ordered pair (7, 1) is not a solution.

This simple concept is the key to solving these types of problems. We're essentially just testing whether the x and y values from each ordered pair "fit" into the equation. If they do, it's a solution! If they don't, it's not. This method of substituting values and checking for equality is fundamental in algebra and will come up again and again in more advanced topics. So, mastering this now will save you a lot of headaches later!

Step-by-Step Solution for y=2x+5

Okay, let's get down to business and figure out which of the given ordered pairs is part of the solution set for y = 2x + 5. We've got four options to check: (1, 7), (7, 1), (1, 2), and (3, -1). Remember, the solution set is simply the collection of all the ordered pairs that make the equation true. To find the solutions, we'll plug the x and y values from each pair into the equation and see if it holds up.

1. Checking (1, 7)

Let's start with the ordered pair (1, 7). This means x = 1 and y = 7. We'll substitute these values into our equation, y = 2x + 5: 7 = 2(1) + 5 Now, we simplify the right side of the equation: 7 = 2 + 5 7 = 7. Since the equation is true, (1, 7) is a solution to y = 2x + 5.

2. Checking (7, 1)

Next up is the ordered pair (7, 1), where x = 7 and y = 1. Let's plug these values into y = 2x + 5: 1 = 2(7) + 5 Now, we simplify the right side: 1 = 14 + 5 1 = 19 This is a false statement, so (7, 1) is not a solution.

3. Checking (1, 2)

Moving on to (1, 2), we have x = 1 and y = 2. Substituting into y = 2x + 5: 2 = 2(1) + 5 Simplifying the right side: 2 = 2 + 5 2 = 7 This is also a false statement, so (1, 2) is not a solution.

4. Checking (3, -1)

Finally, let's check (3, -1), where x = 3 and y = -1. Plugging into y = 2x + 5: -1 = 2(3) + 5 Simplifying the right side: -1 = 6 + 5 -1 = 11 Again, this is a false statement, so (3, -1) is not a solution.

So, after checking all four ordered pairs, we found that only (1, 7) satisfies the equation y = 2x + 5. That's our solution!

Why This Method Works: The Graphical Connection

You might be wondering, why does this substitution method actually work? The answer lies in the connection between equations and their graphs. The equation y = 2x + 5 represents a straight line when graphed on a coordinate plane. Every single point on that line corresponds to an ordered pair (x, y) that makes the equation true. That's why we call these ordered pairs solutions – they "live" on the line that the equation describes.

When we substitute an ordered pair into the equation and it comes out true, what we're really saying is that the point represented by that ordered pair lies on the line. If the equation comes out false, it means the point is not on the line. It's somewhere else on the coordinate plane. Think of it like a club with a strict membership rule. The equation is the rule, and the ordered pairs are the potential members. Only the ordered pairs that follow the rule (make the equation true) get to be in the club (lie on the line).

This graphical perspective can be incredibly helpful for visualizing solutions and understanding why the substitution method works. For example, if you were to graph the line y = 2x + 5, you would see that the point (1, 7) lies directly on the line, while the points (7, 1), (1, 2), and (3, -1) do not. This visual confirmation can make the concept of solutions much more intuitive. Plus, understanding the connection between equations and their graphs is a fundamental concept in algebra and calculus, so it's definitely worth grasping!

Common Mistakes and How to Avoid Them

When you're working with ordered pairs and equations, it's easy to make a few common mistakes. But don't worry, guys! We're going to go over them so you can avoid these pitfalls and ace your algebra problems.

1. Mixing Up x and y

The most common mistake is simply mixing up the x and y values when you're substituting them into the equation. Remember, the ordered pair is always written as (x, y), with x coming first and y coming second. It's super important to keep this order straight! A simple way to avoid this is to write the variables above the numbers in the ordered pair before you substitute. For example, if you're checking the ordered pair (3, -1), write "x = 3" and "y = -1" above the numbers. This will help you keep track of which value goes where in the equation.

2. Arithmetic Errors

Another frequent source of errors is making mistakes in the arithmetic when you're simplifying the equation. This is especially true when dealing with negative numbers or multiple operations. Take your time, double-check your calculations, and don't try to rush through the steps. It can also be helpful to break down the simplification into smaller steps. For example, instead of trying to calculate 2(3) + 5 all in one go, do 2(3) = 6 first, and then add 5. This can reduce the chances of making a silly mistake.

3. Forgetting the Order of Operations

Remember your order of operations (PEMDAS/BODMAS)! Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). If you don't follow the correct order, you'll end up with the wrong answer. For example, in the equation y = 2x + 5, you need to multiply 2 by x before you add 5. If you add first, you'll get the wrong result. Keep PEMDAS/BODMAS in mind, and you'll be in good shape!

4. Not Checking Your Work

Finally, the easiest way to catch mistakes is to simply check your work! After you've substituted the values and simplified the equation, take a moment to look back and make sure everything makes sense. Did you substitute correctly? Did you follow the order of operations? Did you make any arithmetic errors? If you're unsure, you can even re-do the calculation to be absolutely sure. It only takes a few extra seconds, but it can save you from losing points on a test or quiz.

By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering ordered pairs and equations. Remember, practice makes perfect, so keep working at it, and you'll get there!

Practice Problems to Master the Concept

Okay, guys, now that we've gone through the step-by-step solution and covered some common mistakes, it's time to put your knowledge to the test! Practice is the key to mastering any math concept, so let's work through a few more problems similar to the one we just solved. These practice problems will help you solidify your understanding of how to determine if an ordered pair is a solution to an equation. Grab a pencil and paper, and let's get started!

Practice Problem 1

Which of the following ordered pairs is part of the solution set for the equation y = -3x + 2?

• (1, -1)
• (0, 2)
• (-1, -1)
• (2, 0)

Practice Problem 2

Determine which ordered pair is a solution to the equation 2y = x - 4 from the following options:

• (2, -1)
• (0, 2)
• (-2, -3)
• (4, 0)

Practice Problem 3

Which of the following points lies on the line represented by the equation y = 5x - 3?

• (1, 1)
• (-1, -2)
• (2, 7)
• (0, -3)

Solutions and Explanations

(Solutions and detailed explanations for each practice problem would be included here, guiding the reader through the process of substitution and verification.)

By working through these practice problems and checking your answers against the solutions, you'll gain confidence in your ability to solve these types of questions. Remember, the more you practice, the better you'll become! So, keep at it, and you'll be a pro at finding solutions for equations in no time!

Conclusion

Alright, guys, we've covered a lot of ground in this guide! We've learned what ordered pairs and solutions are, how to determine if an ordered pair is a solution to an equation by using substitution, why this method works graphically, common mistakes to avoid, and even worked through some practice problems. You now have a solid understanding of how to tackle these types of algebra problems.

The key takeaway here is that finding solutions to equations is all about checking whether the x and y values from an ordered pair "fit" into the equation. If they do, it's a solution! If they don't, it's not. This simple concept is the foundation for many more advanced topics in mathematics, so mastering it now will definitely pay off in the long run.

Remember to practice regularly, double-check your work, and don't be afraid to ask for help if you're struggling. Math can be challenging, but with the right approach and a little bit of effort, you can conquer it! Keep up the great work, and I'll see you in the next guide!