Solving -x+y=-1 And X+2y=7 With Substitution Method A Step-by-Step Guide

Hey guys! Today, we're diving into the world of solving systems of linear equations using a powerful technique called substitution. If you've ever felt a little lost when faced with two equations and two unknowns, don't worry – this guide will break it down for you. We'll tackle a specific example: solving the system of equations -x + y = -1 and x + 2y = 7. By the end of this, you'll be a substitution pro!

What are Systems of Linear Equations?

Before we jump into the nitty-gritty, let's quickly recap what systems of linear equations are all about. Think of it like this: you have two or more equations, each representing a straight line when graphed. The solution to the system is the point (or points) where these lines intersect. That point satisfies both equations simultaneously. In simpler terms, it's the values for 'x' and 'y' that make both equations true at the same time. There are several methods to solve these systems, and substitution is a key player in our arsenal. Why is substitution so important? Well, it's incredibly versatile and can be applied to a wide range of problems, even when the equations aren't perfectly set up for other methods like elimination. It's also a great way to build your algebraic skills and understand how variables interact within equations. So, whether you're a student tackling algebra homework or just someone who enjoys the challenge of a good mathematical puzzle, mastering substitution is a valuable asset. Now, let's get our hands dirty with our example problem: -x + y = -1 and x + 2y = 7. We'll break down each step so you can see exactly how substitution works in action.

The Substitution Method: A Closer Look

The substitution method, guys, is all about strategically replacing one variable in an equation with an equivalent expression derived from another equation. This might sound a bit abstract right now, but trust me, it'll become crystal clear as we work through our example. The fundamental idea is to reduce a system of two equations with two variables into a single equation with just one variable. This makes the equation solvable, and once you've found the value of that one variable, you can easily backtrack to find the value of the other. It's like solving a puzzle where you find one piece that unlocks the rest! Let's think about the bigger picture for a moment. Why is this method so useful? Well, imagine you have a system of equations where one equation is already neatly solved for one variable in terms of the other. For instance, if you had an equation like y = 3x + 2, substitution would be a natural choice. You could simply substitute the expression '3x + 2' in place of 'y' in the other equation, immediately simplifying the problem. But even if the equations aren't in this perfect form, substitution can still be a powerful tool. We can manipulate one of the equations to isolate a variable and then proceed with the substitution. That's exactly what we'll do with our example problem. So, keep this strategic mindset in mind as we proceed. It's not just about following steps; it's about understanding the underlying logic of substitution and how it helps us unravel the relationships between variables.

Step 1: Isolating a Variable

The first step in our substitution adventure is to isolate one variable in one of the equations. This means we want to get one variable all by itself on one side of the equation, with everything else on the other side. Looking at our system, -x + y = -1 and x + 2y = 7, we have a couple of options. Notice that the 'y' in the first equation and the 'x' in the second equation have coefficients of 1 (or -1 in the case of -x). This is great news because it means isolating them won't involve any messy fractions! For this example, let's choose to isolate 'y' in the first equation, -x + y = -1. To do this, we simply add 'x' to both sides of the equation. This gives us: y = x - 1. Ta-da! We've successfully isolated 'y'. Now, let's pause for a moment and appreciate what we've accomplished. We've transformed the first equation into a new form that expresses 'y' directly in terms of 'x'. This is the key to substitution. We now have an expression, 'x - 1', that we know is equal to 'y'. This expression is our ticket to simplifying the second equation. Think of it like having a secret code that allows us to rewrite the second equation in a more manageable way. Before we move on, it's worth noting that we could have chosen to isolate 'x' in the second equation instead. The beauty of substitution is that there's often more than one way to approach a problem. However, choosing the variable that's easiest to isolate can save you time and effort. In our case, isolating 'y' in the first equation was a relatively straightforward process, which makes it a good choice. So, with our variable isolated and our expression ready, let's move on to the next crucial step: substitution!

Step 2: Substituting the Expression

Alright, guys, we've reached the heart of the substitution method: the substitution itself! We've isolated 'y' in the first equation and found that y = x - 1. Now, we're going to take this expression, 'x - 1', and substitute it in place of 'y' in the second equation. Remember, the goal here is to eliminate one variable and create an equation with just one unknown. Our second equation is x + 2y = 7. So, wherever we see a 'y', we're going to replace it with 'x - 1'. This gives us: x + 2(x - 1) = 7. Now, take a good look at this equation. Notice anything special? That's right! It only contains the variable 'x'. We've successfully transformed our system of two equations into a single equation with one variable. This is a huge step forward. It means we're on the verge of solving for 'x'. But before we can do that, we need to simplify this equation. This involves distributing the '2' across the parentheses and then combining like terms. It's essential to be careful with your algebra here to avoid making mistakes. A small error in this step can throw off your entire solution. So, let's take it one step at a time and make sure we're on the right track. Once we've simplified the equation, we'll have a straightforward linear equation in 'x' that we can easily solve. And once we've found 'x', we'll be well on our way to finding 'y' and completing the solution to our system. So, let's get to it and unleash the power of substitution!

Step 3: Solving for the Remaining Variable (x)

Okay, let's roll up our sleeves and solve for 'x'! We left off with the equation x + 2(x - 1) = 7. Our mission now is to simplify this equation and isolate 'x'. First, we need to distribute the '2' inside the parentheses: x + 2x - 2 = 7. Next, let's combine the 'x' terms: 3x - 2 = 7. Now, we want to get the 'x' term by itself, so we'll add '2' to both sides of the equation: 3x = 9. Finally, to isolate 'x', we divide both sides by '3': x = 3. Hooray! We've found the value of 'x'. It's like discovering a hidden treasure in a mathematical puzzle. But remember, we're not quite done yet. We've only found one piece of the solution. We still need to find the value of 'y'. However, finding 'y' is now a breeze because we know 'x' and we have an equation that relates 'x' and 'y'. This is the beauty of the substitution method. By systematically eliminating one variable, we've made it possible to solve for the other. Now, let's take a moment to appreciate the journey we've taken. We started with a system of two equations that seemed a bit daunting. But by breaking the problem down into manageable steps – isolating a variable, substituting, and simplifying – we've successfully found the value of one of the unknowns. This process highlights the power of algebraic manipulation and the importance of careful, step-by-step reasoning. So, with 'x' in hand, let's move on to the final stage of our adventure: finding 'y' and completing the solution to our system.

Step 4: Finding the Value of y

Now that we've triumphantly found the value of x (x = 3), it's time to hunt down 'y'. Remember that equation we found in step 1 when we isolated 'y'? It was y = x - 1. This equation is our secret weapon for finding 'y' quickly and easily. All we need to do is substitute the value of 'x' we just found (x = 3) into this equation. So, we get: y = 3 - 1. This simplifies to: y = 2. Boom! We've found 'y'. It's like fitting the final piece into a jigsaw puzzle, and the complete picture is now clear. We now know that x = 3 and y = 2. But before we declare victory, it's always a good idea to check our solution. This helps us avoid careless mistakes and ensures that our answer is correct. Checking our solution involves plugging the values of 'x' and 'y' back into the original equations to see if they hold true. If both equations are satisfied, then we know we've found the correct solution. So, let's put our solution to the test and make sure we've conquered this system of equations!

Step 5: Checking the Solution

Alright, guys, let's put on our detective hats and check our solution! We found that x = 3 and y = 2. To make sure these values are correct, we need to plug them back into the original equations: -x + y = -1 and x + 2y = 7. Let's start with the first equation: -x + y = -1. Substitute x = 3 and y = 2: -(3) + 2 = -1. Simplify: -3 + 2 = -1. -1 = -1. This equation holds true! That's a great sign. Now, let's move on to the second equation: x + 2y = 7. Substitute x = 3 and y = 2: 3 + 2(2) = 7. Simplify: 3 + 4 = 7. 7 = 7. This equation also holds true! We've done it! Our solution, x = 3 and y = 2, satisfies both equations in the system. This confirms that we've found the correct solution. It's a wonderful feeling to reach the end of a mathematical journey and know that you've arrived at the right destination. The process of checking our solution is a crucial step in problem-solving. It not only gives us confidence in our answer but also helps us catch any potential errors we might have made along the way. So, remember to always check your work, especially in math! Now that we've successfully solved and checked our solution, let's take a moment to summarize our findings and reflect on the power of the substitution method.

Conclusion: The Solution and the Power of Substitution

So, guys, we've reached the end of our journey to solve the system of equations -x + y = -1 and x + 2y = 7 using the substitution method. After carefully working through each step, we've discovered that the solution is x = 3 and y = 2. This means that the point (3, 2) is the intersection point of the two lines represented by these equations. It's the unique pair of values that satisfies both equations simultaneously. But beyond just finding the solution, we've also explored the power and elegance of the substitution method itself. We've seen how it allows us to systematically transform a system of two equations into a single, solvable equation. We've learned the importance of isolating variables, substituting expressions, simplifying equations, and checking our work. These are valuable skills that will serve you well in your mathematical adventures. The substitution method is not just a technique for solving systems of equations; it's a way of thinking. It's about breaking down complex problems into smaller, more manageable steps. It's about using algebraic manipulation to reveal hidden relationships between variables. And it's about the satisfaction of finding a solution through careful reasoning and logical deduction. So, the next time you encounter a system of equations, remember the power of substitution. It's a tool that can help you unlock the secrets of the mathematical world. And who knows what other mathematical challenges you'll be able to conquer with this newfound skill! Congratulations on mastering the substitution method!