Solving X + 2y = 5 And 3x - 2y = 7 A Step By Step Guide
Hey guys! Are you scratching your heads over systems of equations? Don't worry; you're not alone! Systems of equations can seem daunting, but with the right approach, they become super manageable. Today, we're going to dive deep into solving a specific system: x + 2y = 5 and 3x - 2y = 7. We'll break down each step, making it crystal clear even if you're just starting out with algebra. Think of this as your friendly guide to mastering this type of problem. So, let's get started and turn those equation-solving frowns upside down!
Introduction to Systems of Equations
Before we jump into the nitty-gritty, let's quickly recap what systems of equations are all about. Imagine you've got two or more equations with two or more variables – that's a system of equations. The goal? To find the values of those variables that make all the equations true at the same time. Think of it like a puzzle where each equation is a clue, and you need to find the solution that fits all the clues perfectly. There are several methods to tackle these systems, but we'll focus on the elimination method and the substitution method today. These are the rockstars of equation-solving, and once you've got them down, you'll be unstoppable! Remember, understanding the basics is crucial, so let's make sure we're all on the same page before we dive into our specific problem. Systems of equations pop up everywhere in real life, from calculating costs to planning projects, so mastering them is seriously valuable.
Why Solving Systems of Equations Matters
Okay, so why should you even care about solving systems of equations? Well, imagine you're trying to figure out how many apples and bananas you can buy with a certain amount of money, given their individual prices and your total budget. Boom! That's a system of equations right there. These problems aren't just abstract math exercises; they're tools for tackling real-world scenarios. In fields like engineering, economics, and computer science, systems of equations are used to model complex relationships and make predictions. Whether you're balancing chemical equations, optimizing a business plan, or designing a bridge, the ability to solve these systems is a game-changer. Plus, mastering this skill builds your problem-solving muscles, making you a more effective thinker in all areas of life. So, trust me, learning this stuff is an investment in your future!
Overview of Methods to Solve Systems of Equations
Alright, let's talk methods! There are a few main ways to solve systems of equations, each with its own strengths and weaknesses. We'll be focusing on two of the most popular: the elimination method and the substitution method. The elimination method is like a strategic subtraction game – you manipulate the equations so that one variable cancels out when you add or subtract them. This leaves you with a single variable equation, which is much easier to solve. On the other hand, the substitution method is like a clever swap. You solve one equation for one variable and then substitute that expression into the other equation. Again, this reduces the problem to a single variable equation. There are also graphical methods, where you plot the equations as lines and find their intersection point. And let's not forget matrix methods, which are super powerful for larger systems. But for our problem today, we'll stick with elimination and substitution. Understanding these methods gives you a versatile toolkit for tackling any system of equations that comes your way.
Problem Statement: x + 2y = 5 and 3x - 2y = 7
Okay, let’s get down to business! The system of equations we’re tackling today is:
- x + 2y = 5
- 3x - 2y = 7
This is a classic example of a system of two linear equations with two variables, x and y. Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy both equations simultaneously. In other words, we're looking for the point where these two lines intersect if we were to graph them. This type of problem is incredibly common in algebra, and it’s a fantastic opportunity to practice our equation-solving skills. So, let’s keep this system in mind as we explore different methods to crack the code. Remember, the key is to break it down step by step and not get overwhelmed by the whole thing. You’ve got this!
Understanding the Equations
Before we start crunching numbers, let's make sure we really get what these equations are telling us. Each equation, x + 2y = 5 and 3x - 2y = 7, represents a straight line on a graph. The solutions to each individual equation are all the points that lie on that line. But we're not just looking for any solution; we're looking for the one point that lies on both lines. That point, where the lines intersect, is the solution to the system of equations. Thinking visually can be a huge help here. Imagine two lines crossing each other on a graph – that intersection point is our prize. Understanding this geometric interpretation can make the algebra feel less abstract and more concrete. Plus, it gives you a way to check your work later on – if you have time, you could even sketch the lines and see if your solution looks right!
Goal: Find Values for x and y
So, let's nail down our objective. Our goal is crystal clear: we need to find the values of x and y that make both equations true. It's like we're on a treasure hunt, and x and y are the hidden loot. We can't just guess random numbers and hope they work. We need a systematic approach. That's where the elimination and substitution methods come in. These methods provide us with a roadmap to the solution, guiding us step by step. Think of it like this: we're detectives, and the equations are our clues. By carefully analyzing the clues and using our equation-solving skills, we'll uncover the values of x and y. And when we do, we'll know we've solved the mystery!
Method 1: Elimination Method
Alright, let's dive into the elimination method! This is a super slick technique that works wonders when you can easily cancel out one of the variables. The basic idea is to manipulate the equations so that the coefficients of either x or y are opposites. Then, when you add the equations together, that variable disappears, leaving you with a simpler equation to solve. It’s like magic, but it’s actually just clever algebra! For our system, x + 2y = 5 and 3x - 2y = 7, we’re in luck because the y terms already have opposite coefficients (+2 and -2). This means we're one step closer to cracking the code. So, let’s roll up our sleeves and see how this works in action.
Step 1: Add the Equations
This is where the magic happens! We're going to add the two equations together, term by term. It's like stacking the equations on top of each other and adding the columns. So, we have:
(x + 2y) + (3x - 2y) = 5 + 7
Notice how the +2y and -2y neatly cancel each other out? That's the whole point of the elimination method! We're eliminating one variable to make the equation simpler. Now, let’s simplify the equation. Combining the x terms, we get 4x. The y terms are gone, and on the right side, 5 + 7 equals 12. So, our equation becomes:
4x = 12
See how much simpler that is? We've gone from two equations with two variables to one equation with one variable. That's progress! Now, we're just one step away from finding the value of x.
Step 2: Solve for x
Okay, we've got 4x = 12. Time to solve for x! This is a piece of cake. All we need to do is isolate x by dividing both sides of the equation by 4. Remember, whatever you do to one side of the equation, you have to do to the other to keep things balanced. So, we get:
4x / 4 = 12 / 4
Simplifying this, we find:
x = 3
Woohoo! We've found the value of x. That's half the battle right there. Now we know that x is 3. But we're not done yet. We still need to find the value of y. Don't worry, we're on a roll, and the next step is a breeze.
Step 3: Substitute x into One of the Original Equations
Now that we know x = 3, we can plug this value back into either of the original equations to solve for y. It doesn't matter which equation you choose; you'll get the same answer either way. Let's go with the first equation, x + 2y = 5, because it looks a little simpler. So, we substitute x = 3 into this equation:
3 + 2y = 5
See what we did there? We replaced x with its value, 3. Now we have an equation with just one variable, y. Time to solve for y and complete our mission!
Step 4: Solve for y
Alright, we've got 3 + 2y = 5. Let's isolate y. First, we'll subtract 3 from both sides of the equation to get rid of that pesky 3 on the left:
3 + 2y - 3 = 5 - 3
This simplifies to:
2y = 2
Now, we just need to divide both sides by 2 to solve for y:
2y / 2 = 2 / 2
And that gives us:
y = 1
Yes! We've found the value of y. We now know that y = 1. That means we've cracked the code! We've found the values of x and y that satisfy both equations in our system. Let's take a moment to celebrate our success!
Solution Using Elimination Method
So, after all that awesome equation-solving, we've arrived at our solution! Using the elimination method, we found that:
x = 3 y = 1
This means the solution to our system of equations, x + 2y = 5 and 3x - 2y = 7, is the point (3, 1). This is the point where the two lines represented by these equations intersect. We've successfully found the values of x and y that make both equations true. Give yourself a pat on the back – you've conquered a system of equations! But just to be super sure, let's quickly check our solution to make sure it works in both equations. We don't want any sneaky mistakes ruining our victory!
Method 2: Substitution Method
Now, let's explore another powerful technique: the substitution method. This method is like a clever swap-out. You solve one equation for one variable and then substitute that expression into the other equation. This transforms the problem into a single equation with a single variable, which is much easier to handle. It's like turning a complex puzzle into a simpler one. For our system, x + 2y = 5 and 3x - 2y = 7, we can choose either equation and solve for either variable. The key is to pick the easiest route. So, let’s see how the substitution method works its magic and helps us find those elusive values of x and y.
Step 1: Solve One Equation for One Variable
Okay, let's get started with the substitution method. The first step is to pick one of the equations and solve it for one of the variables. We want to make our lives as easy as possible, so let's look for an equation where a variable has a coefficient of 1. In our system, x + 2y = 5 and 3x - 2y = 7, the first equation, x + 2y = 5, looks like a good candidate. We can easily solve it for x by subtracting 2y from both sides:
x + 2y - 2y = 5 - 2y
This simplifies to:
x = 5 - 2y
Fantastic! We've solved the first equation for x. Now we have an expression for x in terms of y. This is the key to the substitution method. We're going to substitute this expression into the other equation and eliminate x.
Step 2: Substitute the Expression into the Other Equation
Here comes the substitution part! We're going to take the expression we found for x, which is x = 5 - 2y, and substitute it into the other equation, 3x - 2y = 7. It's crucial to substitute into the other equation, not the one we just used. This substitution will eliminate x and give us an equation with just y. So, let’s replace x in the second equation with (5 - 2y):
3(5 - 2y) - 2y = 7
Notice how we've replaced x with the entire expression (5 - 2y). Make sure to use parentheses to keep things organized! Now we have an equation with just y, and we're one step closer to solving for it.
Step 3: Solve for y
Alright, we've got 3(5 - 2y) - 2y = 7. Time to solve for y. First, we need to distribute the 3:
15 - 6y - 2y = 7
Now, let's combine the y terms:
15 - 8y = 7
Next, we'll subtract 15 from both sides:
15 - 8y - 15 = 7 - 15
This gives us:
-8y = -8
Finally, we'll divide both sides by -8 to solve for y:
-8y / -8 = -8 / -8
And that gives us:
y = 1
Yay! We've found the value of y using the substitution method. We know that y = 1. Now we just need to find the value of x. And guess what? We already have an expression for x in terms of y!
Step 4: Substitute y Back to Find x
Now that we know y = 1, we can plug this value back into the expression we found for x in Step 1: x = 5 - 2y. This is the beauty of the substitution method – we already have an equation ready to go. So, let’s substitute y = 1 into this equation:
x = 5 - 2(1)
Simplifying this, we get:
x = 5 - 2
And that gives us:
x = 3
Fantastic! We've found the value of x. We now know that x = 3. That means we've solved the system using the substitution method! We've found the values of x and y that satisfy both equations. High five!
Solution Using Substitution Method
After navigating the substitution method, we've arrived at our solution! We found that:
x = 3 y = 1
Just like with the elimination method, the solution to our system of equations, x + 2y = 5 and 3x - 2y = 7, is the point (3, 1). This confirms that both methods lead us to the same correct answer. We've successfully found the values of x and y that make both equations true using a different approach. You're becoming a system-of-equations-solving superstar! But, just as before, let's take a moment to double-check our solution. It's always a good idea to make sure our answers are spot on.
Verification of Solution
Alright, we've solved our system of equations using two different methods, and both times we got x = 3 and y = 1. That's a pretty good sign that we're on the right track! But to be absolutely sure, we need to verify our solution. This means plugging the values we found for x and y back into the original equations to see if they hold true. It's like the final boss battle in a video game – the last step to claim our victory. So, let’s roll up our sleeves one more time and make sure our solution is rock solid.
Plug the Values of x and y into the Original Equations
Okay, let's plug x = 3 and y = 1 into our original equations:
- x + 2y = 5
- 3x - 2y = 7
For the first equation, we have:
3 + 2(1) = 5 3 + 2 = 5 5 = 5
That checks out! The first equation holds true. Now let's try the second equation:
3(3) - 2(1) = 7 9 - 2 = 7 7 = 7
Woohoo! The second equation also holds true. This means our solution, x = 3 and y = 1, is the real deal. We've officially verified our answer. We can confidently say that we've conquered this system of equations!
Check if the Equations Hold True
We've done the substitution, we've done the simplification, and now we've checked our work. Both equations hold true when we plug in x = 3 and y = 1. This is the ultimate confirmation that we've found the correct solution. Verification is such an important step in math because it catches any little mistakes we might have made along the way. It's like having a safety net – it gives us the confidence to say, “Yes, I got this!” So, always remember to verify your solutions, especially on tests or important assignments. It's the secret weapon of every successful equation solver!
Conclusion
And there you have it, folks! We've successfully solved the system of equations x + 2y = 5 and 3x - 2y = 7 using both the elimination method and the substitution method. We found that x = 3 and y = 1 is the solution that satisfies both equations. We also took the crucial step of verifying our solution to make sure it's spot on. You've now added two more powerful tools to your equation-solving arsenal. Give yourselves a massive pat on the back – you've earned it!
Recap of Solving System of Equations
Let's take a quick victory lap and recap what we've learned. We started with a system of two linear equations and explored two main methods for solving it: elimination and substitution. The elimination method involves manipulating the equations so that one variable cancels out when you add or subtract them. This leaves you with a single variable equation, which you can easily solve. The substitution method, on the other hand, involves solving one equation for one variable and then substituting that expression into the other equation. This also leads to a single variable equation. We saw how both methods led us to the same correct answer. And, most importantly, we emphasized the importance of verifying our solution to ensure accuracy. Solving systems of equations is a fundamental skill in algebra, and you've now got a solid understanding of how to tackle these problems. Bravo!
Importance of Understanding Different Methods
You might be wondering,
