Solving X + 2Y = 7 And X - 2Y = 7 Using Addition, Comparison, And Substitution

Hey guys! Today, we're diving into the fascinating world of systems of equations. Specifically, we're going to tackle the system X + 2Y = 7 and X - 2Y = 7. We'll explore how to solve this using three powerful methods: addition, comparison, and substitution. Trust me, by the end of this article, you'll be a pro at solving these types of problems! So, buckle up and let's get started!

Understanding Systems of Equations

Before we jump into the solutions, let's quickly recap what a system of equations actually is. In simple terms, it's a set of two or more equations that share the same variables. Our goal is to find the values of these variables that satisfy all the equations in the system simultaneously. Think of it as a puzzle where we need to find the perfect values that make all the pieces fit. In our case, we have two equations with two variables, X and Y. Solving a system of equations is a fundamental concept in algebra and has wide-ranging applications in various fields, including engineering, economics, and computer science. Mastering these techniques will not only help you ace your math exams but also equip you with valuable problem-solving skills for real-world scenarios. So, let’s dive deeper into understanding the core concepts before we tackle the specific methods.

The beauty of systems of equations lies in their ability to represent and solve real-world problems. Imagine you're trying to figure out the cost of two different items given their combined prices in different scenarios. Or perhaps you're trying to determine the optimal mix of ingredients for a recipe. These are just a couple of examples where systems of equations can come to the rescue. The key is to translate the word problem into mathematical equations and then apply the techniques we'll be discussing to find the solutions. Understanding the underlying concepts is crucial for this process. We need to grasp how each equation represents a relationship between the variables and how the solution represents the point where these relationships intersect. This intersection is the key to unlocking the answer, as it's the only set of values that satisfies all the equations simultaneously. So, let's keep this in mind as we move forward and explore the different methods for solving systems of equations.

Furthermore, it's important to understand the different types of solutions a system of equations can have. A system can have a unique solution, meaning there's only one set of values for the variables that satisfies all the equations. This is the case we'll be focusing on today. However, a system can also have no solution, meaning there are no values that can satisfy all the equations simultaneously. This happens when the equations represent parallel lines that never intersect. On the other hand, a system can have infinitely many solutions, which occurs when the equations represent the same line or overlapping lines. In this case, any point on the line satisfies all the equations. Recognizing these different possibilities is crucial for interpreting the results of our calculations. It helps us understand the nature of the problem we're solving and whether the solutions we find make sense in the context of the problem. So, as we explore the methods for solving systems of equations, let's also keep in mind the different types of solutions we might encounter and how to interpret them.

Method 1: The Addition Method

The addition method, also known as the elimination method, is a clever technique that involves adding the equations together in a way that eliminates one of the variables. This makes it super easy to solve for the remaining variable. The core idea behind the addition method is to manipulate the equations so that the coefficients of one of the variables are opposites (e.g., +2Y and -2Y). When we add the equations, these terms cancel out, leaving us with a single equation in one variable. This simplified equation is much easier to solve, and once we find the value of that variable, we can substitute it back into one of the original equations to find the value of the other variable. It's like a domino effect – once we knock down one variable, the other one falls into place too.

Let's apply this to our system: X + 2Y = 7 and X - 2Y = 7. Notice that the Y terms already have opposite coefficients (+2Y and -2Y). This is awesome because it means we can skip the manipulation step and go straight to adding the equations. When we add the left-hand sides (X + 2Y) + (X - 2Y), the 2Y and -2Y terms cancel out, leaving us with 2X. When we add the right-hand sides 7 + 7, we get 14. So, our new equation is 2X = 14. Now, it's a piece of cake to solve for X. We simply divide both sides by 2, and we find that X = 7. The magic of the addition method lies in its ability to simplify the system of equations into a single, easily solvable equation. This is particularly useful when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant.

Now that we've found X = 7, we're halfway there! The next step is to substitute this value back into one of the original equations to solve for Y. It doesn't matter which equation we choose; both will lead us to the same answer. Let's choose the first equation, X + 2Y = 7. Substituting X = 7, we get 7 + 2Y = 7. Now, we have a simple equation in Y that we can easily solve. Subtracting 7 from both sides, we get 2Y = 0. Dividing both sides by 2, we find that Y = 0. And there you have it! We've successfully solved the system of equations using the addition method. We found that X = 7 and Y = 0. To be absolutely sure of our answer, we can plug these values back into both original equations and verify that they hold true. This is a good practice to catch any potential errors and ensure that our solution is correct. So, let's double-check: 7 + 2(0) = 7 and 7 - 2(0) = 7. Both equations are satisfied, so we can confidently say that our solution is X = 7 and Y = 0.

Method 2: The Comparison Method

Next up, we have the comparison method. This method is especially handy when one of the variables is already isolated in both equations, or when it's easy to isolate. The basic idea is to express the same variable in terms of the other variable in both equations. Then, since both expressions are equal to the same variable, we can set them equal to each other. This creates a new equation with only one variable, which we can solve easily. Once we find the value of that variable, we can substitute it back into one of the original equations to find the value of the other variable. It's like finding two different paths to the same destination – we can then compare these paths to find our way.

In our system, X + 2Y = 7 and X - 2Y = 7, let's isolate X in both equations. From the first equation, we can subtract 2Y from both sides to get X = 7 - 2Y. From the second equation, we can add 2Y to both sides to get X = 7 + 2Y. Now we have two expressions for X: X = 7 - 2Y and X = 7 + 2Y. Since both expressions are equal to X, we can set them equal to each other: 7 - 2Y = 7 + 2Y. This is the key step in the comparison method – we've created a new equation with only one variable, Y. This equation is much easier to solve than the original system of equations. We've effectively eliminated one variable by comparing the two expressions for X. This technique is particularly useful when the equations are already in a form that makes it easy to isolate one of the variables.

Now, let's solve the equation 7 - 2Y = 7 + 2Y for Y. First, we can subtract 7 from both sides to get -2Y = 2Y. Next, we can add 2Y to both sides to get 0 = 4Y. Finally, we can divide both sides by 4 to find that Y = 0. We've successfully found the value of Y using the comparison method! Now, just like with the addition method, we need to substitute this value back into one of the original equations to find the value of X. Again, it doesn't matter which equation we choose; both will lead us to the same answer. Let's choose the first equation, X + 2Y = 7. Substituting Y = 0, we get X + 2(0) = 7. This simplifies to X = 7. And there you have it! We've solved the system of equations using the comparison method and found that X = 7 and Y = 0. Just to be sure, we can plug these values back into both original equations and verify that they hold true. This is always a good practice to catch any potential errors.

Method 3: The Substitution Method

Last but not least, we have the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. This, like the comparison method, creates a new equation with only one variable, which we can solve. Once we find the value of that variable, we can substitute it back into either of the original equations (or the expression we found earlier) to find the value of the other variable. The substitution method is a powerful tool, especially when one equation is already solved for one variable or when it's easy to solve for one. It's like replacing a piece in a puzzle with its equivalent, which helps us simplify the whole picture.

Let's apply the substitution method to our system: X + 2Y = 7 and X - 2Y = 7. From the first equation, let's solve for X. We can subtract 2Y from both sides to get X = 7 - 2Y. Now, we have an expression for X in terms of Y. This is the key step in the substitution method – we've isolated one variable in terms of the other. Now, we substitute this expression for X into the second equation. The second equation is X - 2Y = 7. Replacing X with (7 - 2Y), we get (7 - 2Y) - 2Y = 7. Notice that we now have an equation with only one variable, Y. This equation is much easier to solve than the original system of equations. We've effectively eliminated one variable by substituting its expression into the other equation. This technique is particularly useful when one equation is already in a form that makes it easy to isolate one of the variables.

Now, let's solve the equation (7 - 2Y) - 2Y = 7 for Y. First, we can simplify the left-hand side by combining like terms: 7 - 4Y = 7. Next, we can subtract 7 from both sides to get -4Y = 0. Finally, we can divide both sides by -4 to find that Y = 0. We've successfully found the value of Y using the substitution method! Now, we need to substitute this value back into one of the equations to find the value of X. We can use either of the original equations, or we can use the expression we found earlier, X = 7 - 2Y. Let's use the expression X = 7 - 2Y, since it's already solved for X. Substituting Y = 0, we get X = 7 - 2(0). This simplifies to X = 7. And there you have it! We've solved the system of equations using the substitution method and found that X = 7 and Y = 0. To be absolutely sure, we can plug these values back into both original equations and verify that they hold true. This is always a good practice to catch any potential errors.

The Solution and Why It's Correct

So, after applying all three methods (addition, comparison, and substitution), we've consistently arrived at the solution X = 7 and Y = 0. This means the correct answer is not among the options A) (1, 3), B) (3, 2), C) (2, 2), or D) (0, 7). There seems to be an error in the provided alternatives. Let's recap why our solution is correct and why the other options are not.

Our solution, X = 7 and Y = 0, satisfies both equations in the system. When we substitute these values into the first equation, X + 2Y = 7, we get 7 + 2(0) = 7, which is true. When we substitute these values into the second equation, X - 2Y = 7, we get 7 - 2(0) = 7, which is also true. This confirms that our solution is indeed correct. Now, let's consider why the other options are incorrect. To be a solution to the system, a pair of values (X, Y) must satisfy both equations. If we substitute the values from any of the options A, B, C, or D into the equations, we'll find that at least one of the equations is not satisfied. For example, let's take option A) (1, 3). Substituting X = 1 and Y = 3 into the first equation, we get 1 + 2(3) = 7, which is true. However, substituting these values into the second equation, we get 1 - 2(3) = 7, which simplifies to -5 = 7, which is false. Therefore, option A is not a solution. We can repeat this process for the other options and find that none of them satisfy both equations. This further validates our solution and highlights the importance of verifying the solution by substituting it back into the original equations.

Conclusion

There you have it! We've successfully solved the system of equations X + 2Y = 7 and X - 2Y = 7 using three different methods: addition, comparison, and substitution. We found that the solution is X = 7 and Y = 0. While the provided alternatives were incorrect, we've demonstrated the process of solving systems of equations and verifying the solution. Remember, practice makes perfect, so keep honing your skills and you'll become a master of systems of equations in no time!