Solving X+y=62 And X-y=9 A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving into a classic algebra problem: solving a system of linear equations. Specifically, we're going to tackle the equations x + y = 62 and x - y = 9. Don't worry if this looks intimidating; we'll break it down step by step, making it super easy to understand. This guide is designed for anyone who wants to brush up on their algebra skills, whether you're a student, a lifelong learner, or just someone who enjoys a good mathematical puzzle. Understanding how to solve systems of equations is a fundamental skill in mathematics and has applications in various fields, including science, engineering, economics, and computer science. By the end of this article, you'll not only be able to solve this particular problem but also have a solid understanding of the methods involved, allowing you to confidently approach similar problems in the future. So, grab a pen and paper, and let's get started on this mathematical adventure together! We'll explore the problem using two primary methods: the substitution method and the elimination method. Both are powerful tools, and understanding them will give you a versatile approach to solving linear equations. Along the way, we'll provide clear explanations, examples, and tips to help you grasp each concept fully. Our goal is to make the process not just understandable but also enjoyable. After all, math can be fun when you see how the pieces fit together. So, let's jump right in and unlock the secrets of these equations!

Understanding the Problem

Before we jump into solving, let's make sure we really understand what we're dealing with. We have two equations:

1. x + y = 62
2. x - y = 9

What these equations tell us is that we're looking for two numbers, which we've called 'x' and 'y'. The first equation says that if you add these two numbers together, you'll get 62. The second equation says that if you subtract 'y' from 'x', you'll get 9. Our mission is to find the specific values of 'x' and 'y' that make both of these statements true at the same time. Think of it like a puzzle where you need to find the right pieces that fit perfectly in two different spots. Each equation represents a relationship between 'x' and 'y', and the solution is the point where these relationships intersect. This is a fundamental concept in algebra, and these types of problems come up all the time in various contexts. For instance, you might use this kind of math to figure out the prices of two items if you know the total cost and the difference in price. Or, you might use it to calculate distances and speeds in physics problems. The key is to recognize that we have two unknowns and two pieces of information, which means we can solve for the unknowns. To make this even clearer, let's visualize what these equations represent graphically. Each equation is a straight line when plotted on a graph. The solution to the system of equations is the point where these two lines intersect. This point gives us the 'x' and 'y' coordinates that satisfy both equations simultaneously. Understanding the problem conceptually is the first step toward solving it. Now that we have a good grasp of what we're trying to achieve, let's move on to the methods we can use to find the solution.

Method 1: The Substitution Method

The substitution method is a powerful technique for solving systems of equations. The main idea behind this method is to solve one equation for one variable and then substitute that expression into the other equation. This way, we reduce the problem to a single equation with one variable, which we can then solve easily. Let's see how this works with our equations:

1. x + y = 62
2. x - y = 9

Step 1: Solve one equation for one variable

Let's take the second equation (x - y = 9) and solve it for 'x'. To do this, we'll add 'y' to both sides of the equation:

x - y + y = 9 + y

x = 9 + y

Now we have 'x' expressed in terms of 'y'. This is a crucial step because we can now substitute this expression into the other equation.

Step 2: Substitute into the other equation

We'll take the expression we found for 'x' (x = 9 + y) and substitute it into the first equation (x + y = 62). This means we'll replace 'x' in the first equation with '9 + y':

(9 + y) + y = 62

Now we have an equation with just one variable, 'y'.

Step 3: Solve the resulting equation

Let's simplify and solve for 'y':

9 + y + y = 62

9 + 2y = 62

Subtract 9 from both sides:

2y = 62 - 9

2y = 53

Divide both sides by 2:

y = 53 / 2

y = 26.5

So, we've found the value of 'y'.

Step 4: Substitute back to find the other variable

Now that we know 'y', we can plug it back into either of the original equations or the expression we found for 'x' (x = 9 + y) to find 'x'. Let's use the expression x = 9 + y:

x = 9 + 26.5

x = 35.5

So, we've found the value of 'x'.

Solution

Therefore, the solution to the system of equations is x = 35.5 and y = 26.5. This means that these values of 'x' and 'y' satisfy both equations simultaneously. To double-check our work, we can plug these values back into the original equations and see if they hold true:

1. 35. 5 + 26.5 = 62 (True)
2. 36. 5 - 26.5 = 9 (True)

Since both equations are true, we've successfully solved the system using the substitution method! This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. Now, let's explore another method to solve the same problem and see how it compares.

Method 2: The Elimination Method

The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. The key idea behind this method is to manipulate the equations so that when you add them together, one of the variables is eliminated. This leaves you with a single equation with one variable, which is much easier to solve. Let's apply this method to our equations:

1. x + y = 62
2. x - y = 9

Step 1: Align the equations

First, make sure the equations are aligned, meaning the 'x' terms are above each other, the 'y' terms are above each other, and the constants are on the same side. Luckily, our equations are already aligned:

x + y = 62 x - y = 9

Step 2: Eliminate one variable

Notice that the 'y' terms have opposite signs (+y and -y). This is perfect for the elimination method! If we add the two equations together, the 'y' terms will cancel out:

(x + y) + (x - y) = 62 + 9

Step 3: Add the equations

Adding the equations term by term, we get:

x + x + y - y = 71

2x = 71

Now we have a simple equation with just 'x'.

Step 4: Solve for the remaining variable

To solve for 'x', divide both sides by 2:

x = 71 / 2

x = 35.5

We've found the value of 'x'!

Step 5: Substitute back to find the other variable

Now that we know 'x', we can substitute it back into either of the original equations to find 'y'. Let's use the first equation, x + y = 62:

37. 5 + y = 62

Subtract 35.5 from both sides:

y = 62 - 35.5

y = 26.5

So, we've found the value of 'y'.

Solution

Therefore, the solution to the system of equations is x = 35.5 and y = 26.5. Notice that this is the same solution we found using the substitution method! This demonstrates that different methods can lead to the same correct answer. To verify our solution, let's plug the values back into the original equations:

1. 38. 5 + 26.5 = 62 (True)
2. 39. 5 - 26.5 = 9 (True)

Both equations hold true, so we've successfully solved the system using the elimination method. The elimination method is particularly useful when the coefficients of one of the variables are the same or easily made the same (or opposite) by multiplying one or both equations by a constant. This allows for a straightforward elimination step. Now that we've explored both the substitution and elimination methods, let's discuss when each method might be more advantageous.

Choosing the Right Method

Now that we've solved the system of equations using both the substitution and elimination methods, you might be wondering: which method is better? Well, the answer is that it depends on the specific problem. Both methods are powerful tools, and understanding when to use each one can make solving systems of equations much more efficient.

When to Use Substitution Method

The substitution method shines when one of the equations can be easily solved for one variable in terms of the other. This is often the case when one of the variables has a coefficient of 1 or -1. For example, in our problem, the equation x - y = 9 could be easily solved for x (x = 9 + y). In such cases, substitution can be a very direct and efficient approach. If you see an equation where a variable is already isolated or can be isolated with minimal effort, substitution might be your best bet. It allows you to quickly express one variable in terms of the other and reduce the system to a single equation.

When to Use Elimination Method

The elimination method is particularly useful when the coefficients of one of the variables are the same or easily made the same (or opposite) by multiplying one or both equations by a constant. In our example, the 'y' terms had opposite coefficients (+y and -y), which made elimination a natural choice. If you notice that adding or subtracting the equations directly will eliminate a variable, or if you can easily manipulate the equations to make this happen, elimination is often the most straightforward method. It avoids the need to solve for one variable in terms of the other, which can sometimes lead to more complex expressions.

General Tips

• Look for the easiest route: Before diving into a method, take a moment to scan the equations and see if one method seems clearly easier than the other.
• Practice makes perfect: The more you practice with both methods, the better you'll become at recognizing which one is most suitable for a given problem.
• Don't be afraid to switch: If you start with one method and find it's getting complicated, you can always switch to the other method.

In summary, both the substitution and elimination methods are valuable tools for solving systems of equations. By understanding their strengths and weaknesses, you can choose the method that best fits the problem at hand and solve it efficiently. Now, let's wrap up our discussion with a quick recap and some final thoughts.

Conclusion

Alright, guys, we've reached the end of our journey through solving the system of equations x + y = 62 and x - y = 9! We've explored two powerful methods: the substitution method and the elimination method. Both methods led us to the same solution: x = 35.5 and y = 26.5, which we verified by plugging the values back into the original equations. We also discussed how to choose the right method based on the structure of the equations. The substitution method is great when one equation can be easily solved for one variable, while the elimination method shines when the coefficients of one variable are the same or easily made the same (or opposite).

Solving systems of equations is a fundamental skill in algebra and has wide-ranging applications in various fields. Whether you're calculating the prices of goods, determining distances and speeds, or modeling complex systems, the ability to solve for multiple unknowns is invaluable. By mastering these techniques, you're not just learning math; you're developing problem-solving skills that will serve you well in many areas of life.

Remember, the key to success in math is practice. The more you work through problems, the more comfortable and confident you'll become. Don't be afraid to try different approaches, make mistakes, and learn from them. Math is a journey, and every problem you solve is a step forward. So, keep practicing, keep exploring, and keep challenging yourself. You've got this!

If you found this guide helpful, share it with your friends and classmates, and let's spread the joy of math together. And if you have any questions or topics you'd like us to cover in the future, let us know in the comments. Thanks for joining us, and happy solving!