Ordered Pair Solutions: Is (x, Y) A Solution?

Hey Plastik Magazine readers! Ever wondered how to check if a specific point is a solution to a linear equation? It's a fundamental concept in mathematics, and we're here to break it down for you in a way that's super easy to understand. We'll use the equation 3x - 4y = -1 as our example and walk through how to determine if ordered pairs like (1, 1), (5, 4), and (-2, -3) are solutions. So, buckle up, and let's dive into the world of ordered pairs and linear equations!

Understanding Ordered Pairs and Equations

Before we jump into the solutions, let's quickly recap what ordered pairs and linear equations are. An ordered pair is a set of two numbers written in the form (x, y), where 'x' represents the horizontal position and 'y' represents the vertical position on a coordinate plane. Think of it like giving directions: you first say how far to go left or right (x), then how far to go up or down (y). A linear equation, on the other hand, is an equation that can be written in the form Ax + By = C, where A, B, and C are constants. When you graph a linear equation, you get a straight line – hence the name 'linear.'

In our case, we have the linear equation 3x - 4y = -1. This equation represents a line on the coordinate plane. Now, the big question is: do the points represented by our ordered pairs (1, 1), (5, 4), and (-2, -3) lie on this line? To find out, we'll use a simple yet powerful technique: substitution. We'll plug in the x and y values from each ordered pair into the equation and see if it holds true. If it does, then the ordered pair is a solution to the equation, meaning it sits right on that line. If not, it's hanging out somewhere else on the coordinate plane. So, let's get started and see which of these ordered pairs are the real deal!

A. Checking the Ordered Pair (1, 1)

Let's start with the ordered pair (1, 1). This means x = 1 and y = 1. To determine if this ordered pair is a solution to the equation 3x - 4y = -1, we substitute these values into the equation. So, wherever we see an 'x', we'll replace it with '1', and wherever we see a 'y', we'll also replace it with '1'. This gives us: 3(1) - 4(1) = -1. Now, we need to simplify this expression and see if it equals -1.

First, we perform the multiplication: 3 * 1 is 3, and 4 * 1 is 4. So, our equation now looks like this: 3 - 4 = -1. Next, we perform the subtraction: 3 minus 4 is -1. Therefore, we have -1 = -1. Look at that! The equation holds true. This means that when we plug in x = 1 and y = 1 into the equation 3x - 4y = -1, we get a true statement. What does this tell us? It tells us that the ordered pair (1, 1) is indeed a solution to the equation. Graphically, this means that the point (1, 1) lies on the line represented by the equation 3x - 4y = -1. One down, two to go! Let's move on to the next ordered pair and see if it also makes the cut.

B. Checking the Ordered Pair (5, 4)

Next up, we have the ordered pair (5, 4). This time, x = 5 and y = 4. We're going to follow the same process as before: substitute these values into the equation 3x - 4y = -1 and see if it holds true. Replacing 'x' with 5 and 'y' with 4, our equation becomes: 3(5) - 4(4) = -1. Time to simplify and see what we get!

First, let's tackle the multiplication: 3 multiplied by 5 is 15, and 4 multiplied by 4 is 16. So, our equation now reads: 15 - 16 = -1. Now, let's do the subtraction: 15 minus 16 equals -1. So, we end up with -1 = -1. Bingo! The equation is true once again. Just like with the first ordered pair, substituting x = 5 and y = 4 into the equation 3x - 4y = -1 gives us a true statement. This means that the ordered pair (5, 4) is also a solution to the equation. On the graph, the point (5, 4) sits perfectly on the line represented by 3x - 4y = -1. Two ordered pairs down, and both are solutions! Let's see if the third one continues the streak.

C. Checking the Ordered Pair (-2, -3)

Alright, last but not least, we have the ordered pair (-2, -3). In this case, x = -2 and y = -3. Let's plug these values into our trusty equation, 3x - 4y = -1, and see what happens. Substituting 'x' with -2 and 'y' with -3, we get: 3(-2) - 4(-3) = -1. Let's break this down step by step.

First, the multiplication: 3 times -2 is -6, and -4 times -3 is +12 (remember, a negative times a negative is a positive). So, our equation now looks like this: -6 + 12 = -1. Next, we perform the addition: -6 plus 12 is 6. This gives us 6 = -1. Hmm, something's not quite right here. 6 is definitely not equal to -1. This means that the equation does not hold true when we substitute x = -2 and y = -3. Therefore, the ordered pair (-2, -3) is not a solution to the equation 3x - 4y = -1. This point does not lie on the line represented by the equation. So, we have a mix of results: two ordered pairs are solutions, and one is not. It's all about plugging in those values and seeing if the equation balances out!

Conclusion: Identifying Solutions to Equations

So, let's recap what we've learned, guys! We set out to determine whether the ordered pairs (1, 1), (5, 4), and (-2, -3) are solutions to the equation 3x - 4y = -1. By substituting the x and y values from each ordered pair into the equation, we were able to figure out which ones fit the bill. We found that (1, 1) and (5, 4) are indeed solutions because they made the equation true. However, (-2, -3) is not a solution as it resulted in an untrue statement.

This process of substitution is a fundamental tool in algebra and is used extensively to solve various mathematical problems. Whether you're dealing with linear equations, quadratic equations, or even more complex systems, knowing how to check if a point is a solution is a crucial skill. It's like having a key that unlocks the mystery of whether a point belongs to a particular line or curve. Remember, math isn't just about numbers and formulas; it's about understanding the relationships between them. Keep practicing, and you'll become a pro at identifying solutions in no time! Stay tuned for more math explorations right here on Plastik Magazine. Peace out!