Solving Linear Equations: Find The Ordered Pair Solution

by Andrew McMorgan 57 views

Hey Plastik Magazine readers! Let's dive into a classic math problem: finding the solution to a system of linear equations. We'll break down how to solve this step-by-step, making it super clear and easy to understand. So, grab your coffee, and let's get started! Our goal is to figure out which ordered pair, represented as (x, y), satisfies both equations in the system. Remember, a system of linear equations is just a set of two or more equations that we are trying to solve simultaneously. The solution to a system of linear equations is the point (x, y) that lies on all the lines represented by the equations. If the lines intersect at one point, there is one solution. If the lines are parallel and do not intersect, there is no solution. If the lines are the same, there are infinitely many solutions. This problem presents us with two linear equations: 3x + y = 1 and 5x + y = 3. Finding the solution means we need to find the values of x and y that make both of these equations true at the same time. There are several ways to solve a system of linear equations, and we'll use a method called elimination. This approach is straightforward and perfect for these types of problems. Essentially, we want to manipulate the equations in a way that allows us to eliminate one of the variables, making it easier to solve for the other. It's like a mathematical puzzle; we're trying to find the missing pieces that fit perfectly into both equations. Understanding this process is crucial not just for this specific problem, but for a wide range of mathematical and real-world applications. Being able to solve systems of linear equations is a fundamental skill that builds a strong foundation in algebra and beyond. This method is not only practical for solving equations but also sharpens your problem-solving skills in general.

Step-by-Step Solution: Elimination Method

Alright, let's get down to the nitty-gritty and solve this thing using the elimination method. Elimination is our best friend here. The goal is to get rid of one of the variables, making it easier to solve for the other. Here's how it works, step-by-step, no sweat:

  1. Set up the equations: We have our two equations:

    • 3x + y = 1
    • 5x + y = 3

    Make sure both equations are lined up nicely. This setup is crucial for the elimination method to work efficiently. We want the x terms, the y terms, and the constants to be aligned vertically.

  2. Eliminate y: Notice that both equations have a y term with the same coefficient (which is 1). This is perfect for elimination! We can subtract one equation from the other to get rid of y. Let's subtract the first equation from the second equation.

    (5x + y) - (3x + y) = 3 - 1

    This simplifies to:

    5x - 3x + y - y = 2

    Which further simplifies to:

    2x = 2

    Awesome! We have eliminated y and have a simple equation with just x.

  3. Solve for x: Now we can easily solve for x. From 2x = 2, we divide both sides by 2:

    x = 1

    So we have our x value: x = 1. High five!

  4. Solve for y: Now that we know x = 1, we can substitute this value back into either of the original equations to solve for y. Let's use the first equation (3x + y = 1):

    3(1) + y = 1

    3 + y = 1

    Subtract 3 from both sides:

    y = -2

    So, y = -2.

  5. Write the Solution as an Ordered Pair: We found that x = 1 and y = -2. The solution is written as an ordered pair (x, y), so our solution is (1, -2). Great job, guys! The solution to the system of linear equations is (1, -2). This means that if we were to graph these two equations, they would intersect at the point (1, -2). It's like finding a secret meeting point for two lines! This point satisfies both equations, making them true simultaneously.

Checking the Solution

We've crunched the numbers and found an answer, but let's make sure we're right. Always a good idea to check your work, right? We'll plug our ordered pair (1, -2) back into both of the original equations to make sure it works. Let's start with the first equation: 3x + y = 1. Substitute x = 1 and y = -2:

3(1) + (-2) = 1 3 - 2 = 1 1 = 1

Great, the first equation checks out! Now let's check the second equation: 5x + y = 3. Substitute x = 1 and y = -2:

5(1) + (-2) = 3 5 - 2 = 3 3 = 3

Awesome, the second equation also checks out. This confirms that our solution (1, -2) is indeed correct. Double-checking is a crucial step in problem-solving. It helps to catch any mistakes early on and ensures that our solution makes sense in the context of the problem. This practice is not only beneficial for math but also for other aspects of life where accuracy is important. This step provides confidence in your solution, ensuring that you're on the right track.

Conclusion: The Final Answer

We've successfully navigated the world of linear equations and emerged victorious! The ordered pair (1, -2) is the solution to the system of equations. To recap: We used the elimination method to solve the system of linear equations 3x + y = 1 and 5x + y = 3. First, we eliminated y by subtracting the first equation from the second. Then, we solved for x, finding that x = 1. After that, we plugged x back into one of the original equations to solve for y, finding that y = -2. We wrote our solution as an ordered pair (1, -2) and then checked our work to make sure our solution was correct. This solution represents the point where the two lines intersect on a graph. This simple, yet powerful technique can be applied to solve various problems in mathematics, science, and even everyday situations. By understanding the fundamentals of solving linear equations, you are equipped with the skills needed to tackle more complex mathematical problems. Mastering these basic concepts is key to building a strong foundation in algebra and beyond. Keep practicing, and you'll become a pro in no time! Keep exploring the wonderful world of mathematics; you never know what discoveries you might make! This understanding isn't just about solving a problem; it's about developing critical thinking and problem-solving skills that are valuable in all aspects of life. So, the next time you encounter a system of linear equations, remember the steps we've covered today, and you'll be well on your way to finding the solution. Keep up the amazing work!