Solving Systems Of Equations: A Step-by-Step Guide

by Tom Lembong 51 views
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Hey guys! Today, we're diving into the world of solving systems of equations. This might sound a bit intimidating at first, but trust me, it's totally manageable. We'll break down the process step by step, making it easy to understand and apply. We're going to tackle the specific system: -x + y = 4 and -x - y = 2. These kinds of problems pop up all the time in math, and getting a handle on them is super useful for everything from algebra to real-world problem-solving. So, let's get started, shall we?

Before we jump into the example, let's quickly recap what a system of equations actually is. Basically, it's a set of two or more equations that we want to solve together. The solution to a system of equations is the set of values for the variables (in our case, x and y) that satisfy all the equations in the system. Think of it like finding a common ground – the values that make everything work out perfectly. There are several methods for solving these systems, but we'll focus on the elimination method here, as it's often the most straightforward for problems like the one we're working on. The elimination method involves manipulating the equations so that when you add or subtract them, one of the variables cancels out, leaving you with a single equation to solve for the remaining variable. Once you find the value of that variable, you can substitute it back into one of the original equations to find the value of the other variable. It's like a math detective game, and it's actually pretty fun once you get the hang of it. We'll walk through each step, ensuring you grasp the concept and how to apply it to solve these equations. We will use the equation to explain. So let's get into the detail. We are going to solve the equation -x + y = 4 and -x - y = 2. Let's see how it works.

Step 1: Setting up the Equations

Alright, let's begin by writing down our equations clearly:

  1. -x + y = 4
  2. -x - y = 2

These are the two equations we need to solve simultaneously. Notice that the 'y' terms have opposite signs. This is a crucial observation because it suggests that the elimination method will be very convenient here. The elimination method works by either adding or subtracting the equations in a way that eliminates one of the variables. In this case, since the 'y' terms have opposite signs (+y and -y), adding the equations will eliminate 'y'. Remember, the goal is to make it easier to solve the problem by getting rid of one variable, so we only need to solve for the other variable. We'll keep this in mind. It's all about making strategic choices to simplify the problem.

Step 2: Eliminating a Variable

Now, let's add the two equations together. When we add the left sides of the equations, we get (-x + y) + (-x - y). And when we add the right sides, we get 4 + 2. Let's write that out:

(-x + y) + (-x - y) = 4 + 2

Simplifying this, we get:

-2x = 6

Notice how the 'y' terms have completely disappeared? That's the magic of the elimination method in action! By adding the equations, we've successfully eliminated 'y', leaving us with a simple equation in terms of 'x' only.

Step 3: Solving for the Remaining Variable

Now we have the equation -2x = 6. This is a piece of cake to solve! To isolate 'x', we need to divide both sides of the equation by -2:

-2x / -2 = 6 / -2

This simplifies to:

x = -3

And there we have it! We've found the value of x. The equation shows x is equal to -3. This is one part of our solution. We're halfway there, guys! We've successfully solved for x, and now we know that x = -3. We will then plug in the value to another equation to find the value of y.

Step 4: Solving for the Other Variable

We've found x, but we still need to find y. To do this, we'll substitute the value of x (-3) into one of the original equations. Let's use the first equation: -x + y = 4. Replace x with -3:

-(-3) + y = 4

This simplifies to:

3 + y = 4

Now, subtract 3 from both sides of the equation to isolate y:

y = 4 - 3

y = 1

Boom! We've found y! y = 1. We've solved for y, completing our solution. Remember that the solution to a system of equations is a pair of values (x, y) that satisfy both equations. Now we know both x and y.

Step 5: Expressing the Solution

Now that we know both x and y, we can express the solution as an ordered pair: (x, y). In our case, x = -3 and y = 1. So, the solution to the system of equations is:

(-3, 1)

This ordered pair represents the point where the two lines represented by the original equations intersect on a graph. This is the only point that satisfies both equations simultaneously. So, our answer is (-3, 1).

Step 6: Verification

It's always a good idea to check your solution to make sure it's correct. To do this, substitute the values of x and y back into both original equations and ensure that both equations hold true. Let's start with the first equation: -x + y = 4. Substituting x = -3 and y = 1, we get: -(-3) + 1 = 3 + 1 = 4. The first equation checks out! Now, let's check the second equation: -x - y = 2. Substituting x = -3 and y = 1, we get: -(-3) - 1 = 3 - 1 = 2. The second equation also checks out. Since both equations hold true with x = -3 and y = 1, we can be confident that our solution is correct. This is called verifying the answer. Congratulations! You've successfully solved the system of equations! Always remember this verification step to avoid any mistakes. Great job!

Additional Tips and Considerations

Solving systems of equations can seem like a puzzle, but with practice, it becomes much easier. Here are a few additional tips:

  • Choose the Right Method: While we used elimination here, substitution is another common method. Choose the method that seems easiest for the specific equations you're working with.
  • Be Careful with Signs: Pay close attention to the positive and negative signs. A small mistake can lead to a wrong answer.
  • Practice, Practice, Practice: The more you practice, the more comfortable you'll become with solving systems of equations. Try different examples to reinforce your understanding.
  • Understand the Concepts: Make sure you understand what a system of equations is and what it means to find a solution. This will help you approach problems with confidence.

More Examples to Explore

Let's consider some other related examples to strengthen your understanding. Here are a few exercises to try on your own:

  1. Solve the system: 2x + y = 5 and x - y = 1
  2. Solve the system: 3x - 2y = 7 and x + y = 4

Remember, the key is to practice and apply the steps we've covered. Keep at it, and you'll become a pro at solving systems of equations!

I hope this guide has helped you understand how to solve systems of equations! Keep practicing, and you'll be acing these problems in no time. If you have any questions, feel free to ask. Keep up the great work, and happy solving! We have learned how to solve the equation, how the solution works, what are the steps, and then we have to verify the answer. We have also considered some extra tips. You got this, guys!