Finding Solutions: Which Ordered Pair Fits The Equation?

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Hey everyone! Let's dive into a common algebra problem: figuring out which ordered pair is a solution to a given equation. In this case, we're tackling the equation 2x + 6y = 10. Don't worry, it's not as scary as it looks! We'll break down the process step-by-step, making it super easy to understand. Ready to find the solution? Let's go!

Understanding the Basics: Ordered Pairs and Equations

Before we jump into the options, let's quickly recap what we're dealing with. An equation like 2x + 6y = 10 represents a relationship between two variables, x and y. The goal is to find values for x and y that make the equation true. These values are written as an ordered pair (x, y). Think of it like a treasure map: the x value tells you how far to go horizontally, and the y value tells you how far to go vertically. The solution to the equation is the point (or points, actually, since there are infinitely many solutions in this case) that satisfies the equation. It's the only one that makes the equation a true statement. A pair of values will be a solution if, when substituted into the equation, they make both sides equal. If the equation holds true, then the ordered pair is indeed a solution. If not, it's not.

So, what does it really mean to be a solution? A solution is simply a set of values for the variables that, when plugged into the equation, make the equation true. For a linear equation in two variables, such as the one we're working with, a solution is any ordered pair (x, y) that lies on the line represented by the equation. A key concept here is substitution. We take the values from our ordered pair and substitute them into the equation in place of the variables. Then we simplify and see if the left side equals the right side. If it does, voila! We've found a solution. If not, it's back to the drawing board.

Now, let's think about how this applies to the equation 2x + 6y = 10. This is a linear equation, and its graph is a straight line. Every point on that line represents a solution to the equation. But how do we find one specific solution among an infinite number of options? We can substitute the x and y values from the answer choices into the equation to see if they satisfy it. If a pair satisfies the equation, it is a solution. Let's start with a general approach that can be used on all the problems of this type. First, understand the question. Identify what is being asked, which is which ordered pair solves the equation. Second, analyze the given equation and the options. Identify the variables and the relationship between them as defined by the equation. Third, Substitute the values from each ordered pair into the equation. For the ordered pair (x, y), replace x with the x-value and y with the y-value in the equation. Fourth, simplify the equation to check if it's true. Perform the arithmetic operations to see if the left-hand side (LHS) of the equation is equal to the right-hand side (RHS). The LHS should be exactly equal to the RHS. Finally, if the equation is true, the ordered pair is a solution. If it's false, the ordered pair is not a solution.

Testing the Options: Which Ordered Pair Works?

Alright, now let's put our knowledge into action and test each of the given options. This is where the real fun begins! We'll take each ordered pair, plug its x and y values into the equation 2x + 6y = 10, and see if it holds true. If the equation is balanced after substitution, the ordered pair is a solution. If not, we keep searching. This method of plugging in values is called substitution, and it is a super important skill in algebra. It helps us check if a value or a set of values meets the criteria of the equation. So, keep your calculators or pencils ready, and let the substitutions begin!

Option A: (3, 1)

Let's start with option A: (3, 1). This means x = 3 and y = 1. Substituting these values into our equation, we get:

  • 2(3) + 6(1) = 10
  • 6 + 6 = 10
  • 12 = 10

Oops! 12 does not equal 10. So, (3, 1) is not a solution. It's important to remember that the equal sign needs to remain balanced, and if you substitute a pair and the balance is off, that particular pair is not a valid solution for your equation. Remember, a solution must make the equation a true statement.

Option B: (2, 1)

Next up, option B: (2, 1). This means x = 2 and y = 1. Substituting these values into the equation, we get:

  • 2(2) + 6(1) = 10
  • 4 + 6 = 10
  • 10 = 10

Bingo! 10 does equal 10. This means that the ordered pair (2, 1) is a solution to the equation 2x + 6y = 10. We've successfully found a solution! It is worth noting here that while one might be able to stop after finding a solution, it's always good practice to check all the options, just to ensure you've found the only correct one. This can help prevent any simple calculation errors.

Option C: (4, 1)

Let's check option C: (4, 1). Here, x = 4 and y = 1. Substituting these values, we get:

  • 2(4) + 6(1) = 10
  • 8 + 6 = 10
  • 14 = 10

Nope! 14 does not equal 10. So, (4, 1) is not a solution.

Option D: (12, 2)

Finally, let's look at option D: (12, 2). This means x = 12 and y = 2. Substituting these values, we have:

  • 2(12) + 6(2) = 10
  • 24 + 12 = 10
  • 36 = 10

Absolutely not! 36 does not equal 10. So, (12, 2) is not a solution. We've gone through all our options, and only one has emerged as a winner.

The Answer Revealed: The Solution to the Equation

After testing all the options, we can confidently say that the ordered pair (2, 1) is the solution to the equation 2x + 6y = 10. This means that when x = 2 and y = 1, the equation holds true. This point lies on the line that represents the equation. Remember, finding solutions to equations is all about finding values that make the equation balanced. Keep practicing with different equations, and you'll become a pro at solving them!

Tips for Success: Mastering Equation Solutions

Here are some helpful tips to help you conquer these types of problems:

  • Double-check your arithmetic: Simple mistakes in calculations can throw off your answer. Take your time and be careful with your math.
  • Show your work: Writing out each step of your substitution and simplification makes it easier to spot errors and understand your process.
  • Understand the concept: Make sure you understand what an ordered pair represents and what it means for it to be a solution to an equation.
  • Practice, practice, practice: The more you practice, the better you'll become at solving these types of problems. Work through various examples to solidify your understanding.
  • Use a calculator: While you should practice doing the math by hand, a calculator can be a lifesaver, especially with more complex equations.

By following these steps and tips, you'll be well on your way to mastering these kinds of problems! Keep practicing, and you'll become a solution-finding superstar in no time! So next time you see an equation, don't be intimidated. Just break it down step by step, and you'll find the solution with ease. Great job, everyone! Keep up the amazing work!