Solving 3x - 2y = 9 And 3x + 2y = -12 A Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of solving systems of equations. Specifically, we're going to tackle the system:
- 3x - 2y = 9
- 3x + 2y = -12
Don't worry if this looks intimidating – we'll break it down step-by-step. By the end of this guide, you'll not only know how to solve this particular system, but you'll also have a solid understanding of the elimination method, which is a super useful tool in your math arsenal. So, grab your pencil and paper, and let's get started!
Understanding Systems of Equations
Before we jump into the solution, let's quickly recap what a system of equations actually is. Simply put, it's a set of two or more equations that share the same variables. In our case, we have two equations, and both of them involve the variables 'x' and 'y'. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Think of it like finding the secret coordinates (x, y) that satisfy both equations. There are several methods to solve these systems, such as substitution, graphing, and elimination. Today, we're focusing on the elimination method because it's particularly efficient for this specific problem. The core idea behind the elimination method is to manipulate the equations in such a way that when we add (or subtract) them, one of the variables cancels out. This leaves us with a single equation in one variable, which is much easier to solve. 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. This method is especially handy when the coefficients of one of the variables are opposites or easily made opposites, as you'll see in our example. So, buckle up, because we're about to put this method into action and find the solution to our system of equations!
The Elimination Method: Our Strategic Approach
Now, let's talk strategy. The elimination method is our weapon of choice for this problem, and here's why: Notice that the 'y' terms in our equations have opposite coefficients (-2 and +2). This is a huge advantage because it means that if we simply add the two equations together, the 'y' terms will cancel each other out, leaving us with an equation in just 'x'. This neat trick is the heart of the elimination method, making it a powerful tool when you spot opposite coefficients or coefficients that can easily be made opposite. The key idea is to strategically manipulate the equations to eliminate one variable, making the problem much simpler to solve. Once we've eliminated 'y' and found the value of 'x', we can then substitute that value back into either of the original equations to solve for 'y'. It's like a mathematical domino effect – eliminate one variable, and the rest fall into place! This method is not only efficient but also quite elegant, offering a clear and direct path to the solution. We'll walk through each step in detail, so you can see exactly how it works. Get ready to witness the magic of the elimination method as we tackle our system of equations head-on!
Step 1: Adding the Equations
Alright, let's get our hands dirty! This is where the magic of elimination really starts to happen. Remember, our goal is to add the two equations together in a way that eliminates one of the variables. Looking at our system:
- 3x - 2y = 9
- 3x + 2y = -12
We can see that the '-2y' in the first equation and the '+2y' in the second equation are perfectly poised to cancel each other out. So, let's add the equations term by term. We'll add the 'x' terms, the 'y' terms, and the constant terms separately. This gives us:
(3x + 3x) + (-2y + 2y) = (9 + (-12))
Now, let's simplify this expression. 3x plus 3x is 6x. -2y plus 2y cancels out completely, leaving us with zero. And 9 plus -12 is -3. So, our equation becomes:
6x = -3
Boom! Just like that, we've eliminated 'y' and are left with a single equation involving only 'x'. This is a huge step forward, as we've reduced the complexity of the problem significantly. By carefully adding the equations, we've created a simpler equation that we can easily solve for 'x'. This step perfectly illustrates the power of the elimination method, transforming a system of two equations into a single, manageable equation. Next up, we'll solve this equation to find the value of 'x'.
Step 2: Solving for x
Great job on making it this far, guys! Now that we've successfully eliminated 'y', we're left with the simple equation:
6x = -3
This is a piece of cake to solve! Our goal here is to isolate 'x' on one side of the equation. To do this, we need to get rid of the '6' that's multiplying 'x'. The opposite of multiplication is division, so we'll divide both sides of the equation by 6. This is a crucial step – remember, whatever you do to one side of the equation, you must do to the other side to keep the equation balanced.
So, dividing both sides by 6, we get:
(6x) / 6 = (-3) / 6
On the left side, the 6s cancel each other out, leaving us with just 'x'. On the right side, -3 divided by 6 simplifies to -1/2. So, our solution for 'x' is:
x = -1/2
Or, if you prefer decimals:
x = -0.5
Fantastic! We've found the value of 'x'. This is a significant milestone in solving our system of equations. But don't celebrate just yet – we're only halfway there. We still need to find the value of 'y'. But now that we know 'x', we can use that information to find 'y' relatively easily. In the next step, we'll substitute this value of 'x' back into one of the original equations to solve for 'y'. Keep up the awesome work!
Step 3: Substituting x to Find y
Alright, we've nailed down the value of 'x' – it's -1/2. Now, the next step is to find 'y'. To do this, we'll use a technique called substitution. The idea is simple: we'll take the value of 'x' that we just found and plug it into one of the original equations. It doesn't matter which equation we choose, as both will give us the same answer for 'y'. For the sake of simplicity, let's choose the first equation:
3x - 2y = 9
Now, we'll replace 'x' with its value, -1/2:
3 * (-1/2) - 2y = 9
This simplifies to:
-3/2 - 2y = 9
Now, we have an equation with just one variable, 'y', which we can solve. First, let's get rid of the fraction. We can add 3/2 to both sides of the equation:
-3/2 + 3/2 - 2y = 9 + 3/2
This simplifies to:
-2y = 9 + 3/2
To add 9 and 3/2, we need a common denominator. We can rewrite 9 as 18/2. So, we have:
-2y = 18/2 + 3/2
-2y = 21/2
Now, to isolate 'y', we'll divide both sides of the equation by -2. Remember that dividing by a negative number is the same as multiplying by its reciprocal, so we'll multiply both sides by -1/2:
(-2y) * (-1/2) = (21/2) * (-1/2)
This simplifies to:
y = -21/4
Or, as a decimal:
y = -5.25
Excellent! We've found the value of 'y'. Now we have both 'x' and 'y', which means we've solved the system of equations!
Step 4: Verifying the Solution
Hold on there, champs! We've found our potential solution: x = -1/2 and y = -21/4. But before we declare victory, it's crucial to verify our solution. This means plugging our values for 'x' and 'y' back into the original equations to make sure they hold true. This step is like a final safety check, ensuring that we haven't made any mistakes along the way. It's a small investment of time that can save you from incorrect answers and frustration. So, let's take our values and substitute them into both of the original equations:
Equation 1: 3x - 2y = 9
Let's plug in x = -1/2 and y = -21/4:
3 * (-1/2) - 2 * (-21/4) = 9
Simplifying, we get:
-3/2 + 42/4 = 9
To add these fractions, we need a common denominator. Let's rewrite -3/2 as -6/4:
-6/4 + 42/4 = 9
36/4 = 9
9 = 9
Equation 2: 3x + 2y = -12
Now, let's do the same for the second equation:
3 * (-1/2) + 2 * (-21/4) = -12
Simplifying, we get:
-3/2 - 42/4 = -12
Again, let's rewrite -3/2 as -6/4:
-6/4 - 42/4 = -12
-48/4 = -12
-12 = -12
Both equations hold true! This means our solution is correct. We've successfully navigated the system of equations and found the values of 'x' and 'y' that satisfy both equations. Give yourselves a pat on the back – you've earned it!
The Final Solution
Drumroll, please! After all our hard work, we've arrived at the final destination: the solution to the system of equations is:
- x = -1/2 (or -0.5)
- y = -21/4 (or -5.25)
We can express this solution as an ordered pair (x, y): (-1/2, -21/4) or (-0.5, -5.25). This ordered pair represents the point where the two lines represented by our equations intersect on a graph. It's the one and only point that satisfies both equations simultaneously. We not only found the solution but also verified it, ensuring that our answer is correct. This is a testament to the power of the elimination method and your perseverance in working through the steps. Solving systems of equations is a fundamental skill in algebra and has applications in various fields, from science and engineering to economics and computer science. By mastering this skill, you've added another valuable tool to your mathematical toolkit. So, congratulations on conquering this challenge! You've shown that you can tackle complex problems with confidence and precision. Now, go forth and solve more equations!
Conclusion: Mastering Systems of Equations
Woohoo! We did it, guys! We successfully solved the system of equations 3x - 2y = 9 and 3x + 2y = -12 using the elimination method. We walked through each step, from understanding the problem to verifying our solution. We learned how to strategically add equations to eliminate variables, how to solve for the remaining variable, and how to substitute that value back into the original equations to find the other variable. We also emphasized the importance of verifying our solution to ensure accuracy. This journey through solving this system of equations has provided you with valuable skills and insights into the world of algebra. You've not only learned a specific method but also developed a problem-solving mindset that can be applied to other mathematical challenges. Remember, math is not just about memorizing formulas; it's about understanding concepts and applying them creatively. By practicing and persevering, you can master any mathematical concept. So, keep exploring, keep learning, and keep solving! The world of mathematics is full of fascinating puzzles waiting to be discovered. And who knows, maybe you'll be the one to solve the next big problem! Keep up the amazing work, and I'll see you in the next math adventure!
