Solve 3x - 2y = 1 & 2x + 4y = 12: A Step-by-Step Guide

Hey guys! Ever find yourself staring at a system of equations and feeling totally lost? Don't worry, we've all been there. Systems of equations might seem intimidating at first, but with a little know-how, they're actually pretty straightforward to solve. In this article, we're going to break down a common type of problem: solving a system of two linear equations with two variables. We'll use the example of 3x - 2y = 1 and 2x + 4y = 12 to illustrate the process. So, grab a pencil and paper, and let's dive in!

Understanding Systems of Equations

Before we jump into solving, let's quickly recap what a system of equations actually is. Basically, it's a set of two or more equations that involve the same variables. Our goal is to find values for those variables that satisfy all the equations in the system simultaneously. Think of it like finding the secret code that unlocks all the doors at once! In our case, we have two equations, each with two variables, x and y. This means we're looking for a pair of numbers (an x value and a y value) that make both equations true. There are several methods we can use to solve systems of equations, but we'll focus on two popular ones: the substitution method and the elimination method. We'll walk through both, so you can choose the one that clicks best with your brain. These methods aren't just abstract math concepts, guys. They're tools that are used in countless real-world applications, from engineering and physics to economics and computer science. So, mastering these techniques is a valuable skill that can open doors to all sorts of exciting fields. The beauty of systems of equations is that they allow us to model complex situations with multiple unknowns. By setting up the equations correctly, we can use mathematical techniques to find the solutions, even when the problems seem incredibly complicated at first glance. For example, imagine you're trying to figure out how much to charge for two different products you're selling, given certain constraints on your costs and profits. A system of equations can be the perfect way to solve this kind of business problem. Or, suppose you're designing a bridge and need to calculate the forces acting on different parts of the structure. Systems of equations come in handy again! The key is to translate the real-world situation into a set of mathematical relationships, and then use the techniques we'll be exploring to find the answers.

Method 1: The Substitution Method

The substitution method is all about isolating one variable in one equation and then substituting that expression into the other equation. This might sound like a mouthful, but it's easier than it seems! Let's take our system:

• 3x - 2y = 1
• 2x + 4y = 12

Our first step is to choose one equation and one variable to isolate. Looking at our equations, it seems easiest to isolate x in the first equation. So, let's do that. We'll start by adding 2y to both sides of the equation:

3x = 1 + 2y

Then, we'll divide both sides by 3 to get x by itself:

x = (1 + 2y) / 3

Great! We've got an expression for x in terms of y. Now comes the substitution part. We're going to take this expression and plug it into the other equation (the second one in our system). This is crucial because we want to combine the information from both equations. So, we replace the x in the second equation with (1 + 2y) / 3:

2 * ((1 + 2y) / 3) + 4y = 12

Now we have one equation with only one variable, y. This is something we can solve! Let's simplify the equation. First, we'll distribute the 2:

(2 + 4y) / 3 + 4y = 12

To get rid of the fraction, we can multiply both sides of the equation by 3:

2 + 4y + 12y = 36

Combine the y terms:

16y + 2 = 36

Subtract 2 from both sides:

16y = 34

Finally, divide both sides by 16 to solve for y:

y = 34 / 16 = 17 / 8

Woohoo! We've found the value of y. Now, to find x, we simply substitute this value of y back into either of our original equations (or the expression we derived for x). Let's use the expression we found earlier: x = (1 + 2y) / 3. Plugging in y = 17/8, we get:

x = (1 + 2 * (17/8)) / 3

x = (1 + 17/4) / 3

x = (21/4) / 3

x = 7 / 4

So, we've found our solution! x = 7/4 and y = 17/8. This means the point (7/4, 17/8) is the solution to our system of equations. It's where the lines represented by these equations intersect on a graph.

Method 2: The Elimination Method

The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. The basic idea is to manipulate the equations so that when you add them together, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve. Let's see how it works with our system:

• 3x - 2y = 1
• 2x + 4y = 12

Notice that the coefficients of y in the two equations are -2 and 4. If we could somehow make those coefficients opposites (like -4 and 4), then when we add the equations together, the y terms would disappear. So, let's multiply the first equation by 2. This will give us a -4y term:

2 * (3x - 2y) = 2 * 1

Which simplifies to:

6x - 4y = 2

Now we have a modified system:

• 6x - 4y = 2
• 2x + 4y = 12

See how the y coefficients are now -4 and 4? Perfect! Now, we add the two equations together. This means we add the left-hand sides and the right-hand sides separately:

(6x - 4y) + (2x + 4y) = 2 + 12

Simplifying, we get:

8x = 14

Now we have a simple equation with just x. Divide both sides by 8 to solve for x:

x = 14 / 8 = 7 / 4

Look familiar? We got the same value for x as we did with the substitution method! This is a good sign – it means we're on the right track. Now, to find y, we can substitute this value of x back into either of our original equations. Let's use the second equation, 2x + 4y = 12:

2 * (7/4) + 4y = 12

Simplifying:

7/2 + 4y = 12

Subtract 7/2 from both sides:

4y = 12 - 7/2

4y = 17 / 2

Divide both sides by 4:

y = (17/2) / 4 = 17 / 8

And there you have it! We found y = 17/8, which is the same value we got using the substitution method. So, once again, our solution is x = 7/4 and y = 17/8. The elimination method can be particularly useful when the coefficients of one of the variables are already opposites or are easy to make opposites by multiplying one or both equations by a constant. It can often be a quicker method than substitution in those cases. However, both methods are perfectly valid, and it's a matter of personal preference which one you choose to use.

Verifying the Solution

Okay, guys, we've got our solution: x = 7/4 and y = 17/8. But before we declare victory, it's always a good idea to check our answer. How do we do that? Simple! We plug our values for x and y back into the original equations and see if they hold true. This is a crucial step to catch any potential errors we might have made along the way. If the equations are satisfied, then we can be confident that our solution is correct. If not, then we know we need to go back and double-check our work. So, let's plug our values into the first equation, 3x - 2y = 1:

3 * (7/4) - 2 * (17/8) = 1

Simplifying:

21/4 - 17/4 = 1

4/4 = 1

1 = 1

Great! The first equation checks out. Now let's try the second equation, 2x + 4y = 12:

2 * (7/4) + 4 * (17/8) = 12

Simplifying:

7/2 + 17/2 = 12

24/2 = 12

12 = 12

Awesome! The second equation also checks out. Since our values for x and y satisfy both equations, we can confidently say that our solution is correct. This verification step is like a final exam for our answer, and it's always worth taking the time to do it. It not only gives us peace of mind but also helps us develop good problem-solving habits.

Conclusion

So there you have it! We've successfully solved the system of equations 3x - 2y = 1 and 2x + 4y = 12 using both the substitution method and the elimination method. We found that x = 7/4 and y = 17/8. More importantly, we've walked through the logic and steps involved in each method, so you can tackle similar problems with confidence. Remember, practice makes perfect! The more you work with systems of equations, the more comfortable you'll become with the techniques. Don't be afraid to try different approaches and see what works best for you. And remember to always verify your solutions to ensure accuracy. Solving systems of equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By mastering these techniques, you're not just learning math; you're developing valuable problem-solving skills that will serve you well in many aspects of life. So, keep practicing, keep exploring, and keep having fun with math! You've got this, guys!