Solving Linear Equations: A Step-by-Step Guide
Hey guys! Let's dive into solving a system of linear equations. Specifically, we're gonna figure out the solution set for the following: 4x - y = 5 and 3x - y = 3. Don't worry, it's not as scary as it sounds. We'll break it down into easy, digestible steps. Understanding how to solve these equations is a fundamental skill in algebra, and it's super useful for all sorts of real-world problems. Whether you're balancing a budget, figuring out the best deal on a purchase, or even designing a basic computer program, linear equations are everywhere. So, grab your pencils and let's get started! We'll explore a couple of different methods to get to the answer, making sure you have a solid grasp of the concepts.
The Elimination Method: Wiping Out Variables
First off, let's tackle this problem using the elimination method. The elimination method is all about getting rid of one of the variables (either x or y) by manipulating the equations. This makes it easier to solve for the remaining variable. In our case, the equations are 4x - y = 5 and 3x - y = 3. Notice how both equations have a -y term. This is a good sign because it means we can easily eliminate y. To do this, we're going to subtract the second equation from the first. Think of it like this: If we subtract equal quantities from both sides of an equation, the equation remains balanced. We're essentially subtracting the entire left side of the second equation from the left side of the first equation, and we're doing the same for the right sides. This way, we maintain the equality. Subtracting the equations, we get:
(4x - y) - (3x - y) = 5 - 3
Now, let's simplify. 4x - 3x is just x, and -y - (-y) becomes -y + y, which is 0. On the right side, 5 - 3 is 2. So, we're left with:
x = 2
Awesome! We've solved for x. But we're not done yet. We still need to find the value of y. This is where substitution comes in. We know that x = 2, so we can substitute this value back into either of the original equations to solve for y. Let's use the first equation, 4x - y = 5. Replacing x with 2, we get:
4(2) - y = 5
Which simplifies to:
8 - y = 5
Now, let's isolate y. Subtract 8 from both sides:
-y = -3
Finally, multiply both sides by -1 to get the value of y:
y = 3
Therefore, the solution to the system of equations is x = 2 and y = 3. We've successfully used the elimination method to solve the problem!
The Substitution Method: Plugging and Solving
Now, let's solve the same system of equations using the substitution method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Again, the equations are 4x - y = 5 and 3x - y = 3. This time, let's solve the second equation for y. This means we want to isolate y on one side of the equation. Starting with 3x - y = 3, add y to both sides and subtract 3 from both sides. We get:
3x - 3 = y
Or, written the other way around:
y = 3x - 3
Now we know what y is equal to: 3x - 3. We'll substitute this expression for y into the first equation, 4x - y = 5. Replacing y with 3x - 3, we get:
4x - (3x - 3) = 5
Careful with the minus sign here! Distribute the negative sign to both terms inside the parentheses:
4x - 3x + 3 = 5
Simplify the left side:
x + 3 = 5
Subtract 3 from both sides to solve for x:
x = 2
We found that x = 2. Excellent! Now, substitute this value back into the expression we found for y: y = 3x - 3. Replace x with 2:
y = 3(2) - 3
Which simplifies to:
y = 6 - 3
y = 3
Again, we get the solution x = 2 and y = 3. Both the elimination and substitution methods lead to the same solution, confirming our answer. Pretty cool, right? You can choose whichever method feels easier for you, or the one that's better suited to the specific equations you're working with.
Verification: Checking Your Work
It's always a good idea to check your solution. Verifying your solution ensures that your values for x and y actually satisfy both of the original equations. This is a crucial step to avoid careless errors. Let's substitute x = 2 and y = 3 into both equations:
Equation 1: 4x - y = 5
4(2) - 3 = 5
8 - 3 = 5
5 = 5
The first equation checks out! Now, let's check the second equation:
Equation 2: 3x - y = 3
3(2) - 3 = 3
6 - 3 = 3
3 = 3
The second equation also checks out. Since both equations are true with our values for x and y, we can be confident that our solution, x = 2 and y = 3, is correct.
Real-World Applications
So, why does any of this matter? Well, linear equations and their solutions are incredibly useful in a bunch of real-world scenarios. Let's look at a few examples. Imagine you're running a small business and you want to calculate your profit. You have fixed costs (like rent) and variable costs (like the cost of materials). Linear equations can help you model your costs and revenue. By setting up a system of equations, you can determine your break-even point – the point where your revenue equals your costs. Or, let's say you're planning a trip and need to compare different transportation options. Each option might have a fixed cost (like a plane ticket) and a cost per mile (like gas for a car). You can create equations to represent the total cost of each option. Solving the system of equations will tell you when the costs are equal. This helps you decide which option is more economical for your trip. Furthermore, in computer science and programming, linear equations are fundamental to various algorithms and data structures. For example, they're used in machine learning models, graphics rendering, and optimization problems. Whether you're into business, travel, or technology, the ability to solve linear equations will come in handy. It's a fundamental mathematical skill that opens doors to understanding and solving complex problems.
Tips for Success
Alright, here are some tips for success when solving linear equations:
- Practice Regularly: The more you practice, the better you'll get. Work through various examples, and don't be afraid to make mistakes. That's how we learn!
- Double-Check Your Work: Always verify your solution by substituting the values of x and y back into the original equations. This helps catch any errors you might have made.
- Choose the Right Method: Sometimes, one method (elimination or substitution) is easier to use than the other, depending on the equations. Look at the equations carefully and choose the method that seems most straightforward.
- Pay Attention to Detail: Be careful with signs (positive and negative) and distribution. These are common areas where mistakes can happen.
- Break Down the Problem: If the equations seem overwhelming, break them down into smaller, manageable steps. This will make the process less intimidating.
By following these tips and practicing consistently, you'll become a pro at solving linear equations in no time! Keep going, and you'll build a solid foundation in algebra, setting you up for success in more advanced mathematical concepts.
Conclusion: You've Got This!
So there you have it, guys! We've successfully solved a system of linear equations using both the elimination and substitution methods. We've verified our answer and explored some real-world applications and, more importantly, learned some tips for how to solve any linear equation. Remember that the key to success is practice and understanding the underlying concepts. Math can seem daunting at times, but breaking it down into smaller, manageable steps makes it much easier to grasp. Understanding the fundamentals of linear equations gives you a significant advantage in various fields. Keep practicing and keep asking questions. You've totally got this! Feel free to ask more questions. Good luck, and happy solving!