Solving Linear Equations Finding Ordered Pair Solutions
Hey guys! Today, we're diving into the world of linear equations and figuring out which ordered pairs are the real MVPs—the ones that actually solve the equation. We've got a fun problem to tackle, so let's jump right in!
The Challenge: Finding the Right Fit
Our mission, should we choose to accept it (and we totally do!), is to determine which ordered pair(s) satisfy the equation:
3x + 3y = -x + 5y
We're given a few options:
A. Only (1, 2) B. Only (2, 4) C. Both (1, 2) and (2, 4) D. Neither
So, how do we crack this code? It's all about plugging in those x and y values and seeing if they make the equation true. Let's get to it!
Step-by-Step Solution
To figure out which ordered pair is a solution to the equation, we need to test each ordered pair by substituting the x and y values into the equation and seeing if both sides are equal. This is a crucial concept in algebra, and mastering it will help you solve more complex problems down the road. Remember, an ordered pair is written in the format (x, y), where the first number represents the x-coordinate and the second number represents the y-coordinate. When we substitute these values into the equation, we're essentially checking if the left-hand side (LHS) of the equation is equal to the right-hand side (RHS). If they are equal, then the ordered pair is a solution; otherwise, it's not. This process might seem straightforward, but it's important to be meticulous and double-check your work to avoid making mistakes. Math, after all, is a precise language, and even a small error can lead to a wrong answer. So, let's take our time, be careful, and make sure we get it right! To make things super clear, we'll go through each ordered pair one by one, showing you exactly how the substitution and simplification work. This way, you'll not only get the answer to this particular problem but also gain a solid understanding of the method, which you can apply to countless other equations and ordered pairs. So, let's roll up our sleeves and get started! Remember, math is not just about finding the right answer; it's about the journey of understanding and the satisfaction of cracking the code. So, let's enjoy the process and learn something new together.
Testing Option A: (1, 2)
Let's start with the ordered pair (1, 2). We'll plug x = 1 and y = 2 into our equation:
3(1) + 3(2) = -1 + 5(2)
Now, we simplify each side:
3 + 6 = -1 + 10
9 = 9
Woohoo! The equation holds true. This means (1, 2) is a solution. But, hold your horses, we need to check the other ordered pair as well.
Testing Option B: (2, 4)
Next up, we've got the ordered pair (2, 4). Let's plug x = 2 and y = 4 into the equation:
3(2) + 3(4) = -2 + 5(4)
Simplifying again:
6 + 12 = -2 + 20
18 = 18
Boom! This ordered pair also makes the equation true. So, (2, 4) is also a solution.
The Verdict
We've tested both ordered pairs, and guess what? Both (1, 2) and (2, 4) are solutions to the equation. That means the correct answer is:
C. Both (1, 2) and (2, 4)
Key Takeaways and Further Exploration
This problem highlights a fundamental concept in algebra: how to verify solutions to equations. We've seen how substituting values into an equation can confirm whether an ordered pair is a solution. But, there's more to this than meets the eye. Understanding the nature of solutions opens doors to a broader understanding of linear equations and their graphical representations. Each solution we find corresponds to a point that lies on the line represented by the equation. This connection between algebra and geometry is a powerful one, and exploring it further can deepen your mathematical intuition.
Visualizing Solutions on a Graph
Imagine plotting the line represented by our equation, 3x + 3y = -x + 5y. If we were to graph this line, we would see that the points (1, 2) and (2, 4) both lie directly on the line. This is a visual confirmation that these ordered pairs are indeed solutions. The line itself represents the infinite set of solutions to the equation, with each point on the line corresponding to an ordered pair that satisfies the equation. This visual representation can be incredibly helpful for understanding the nature of solutions and the relationships between variables in an equation. For example, if we were to look at the graph, we could easily identify other points that lie on the line and, therefore, are also solutions. This is just one of the many ways that graphing can enhance our understanding of algebra.
The Importance of Algebraic Manipulation
Before we started substituting values, we could have simplified the equation to make it easier to work with. This is a common strategy in algebra, and it's often essential for solving more complex problems. In our case, we could combine like terms to rewrite the equation in a more standard form, such as slope-intercept form (y = mx + b). This form makes it easier to visualize the line and identify key features, such as the slope and y-intercept. It also provides another way to verify our solutions. By plugging the x and y values of our ordered pairs into the simplified equation, we can confirm that they still satisfy the equation. This reinforces the idea that algebraic manipulation doesn't change the underlying solutions; it simply presents the equation in a different, often more convenient, form. Learning how to manipulate equations effectively is a critical skill in algebra, and it's something you'll use again and again as you progress in your mathematical journey.
Beyond Two Points: Infinite Solutions
It's important to remember that a linear equation typically has an infinite number of solutions. We've found two solutions in this problem, but there are countless others. Each of these solutions represents a point on the line, and since a line extends infinitely in both directions, there are infinitely many points on it. This is a fundamental concept in algebra, and it distinguishes linear equations from other types of equations that may have a finite number of solutions. The idea of infinite solutions can seem abstract, but it's crucial for understanding the power and scope of linear equations. They're not just about finding a couple of numbers that work; they're about describing a relationship between variables that holds true across an entire spectrum of values. This is what makes linear equations so versatile and applicable to a wide range of real-world situations, from modeling population growth to predicting financial trends. The fact that there are infinitely many solutions to explore is one of the exciting things about algebra, and it's what keeps mathematicians engaged and inspired.
Wrapping Up
So, there you have it! We successfully navigated the world of ordered pairs and linear equations, proving that both (1, 2) and (2, 4) are solutions to our equation. Remember, guys, practice makes perfect, so keep plugging away at those problems, and you'll become math whizzes in no time! Keep exploring, keep questioning, and most importantly, keep having fun with math!
