Solving 4x + 2y = -2 And -9x - 5y = 1 Using Elimination Method

Hey guys! Ever found yourself staring at a pair of equations and feeling totally lost? Don't worry, we've all been there! Today, we're going to break down how to solve systems of linear equations using a method called elimination. We'll tackle a specific example: 4x + 2y = -2 and -9x - 5y = 1. Trust me, by the end of this, you'll be solving these like a pro!

Understanding Systems of Linear Equations

Before we dive into the solution, let’s make sure we’re all on the same page. Systems of linear equations, at their core, are just sets of two or more linear equations containing the same variables. Think of it like this: you have multiple pieces of information (equations) about the same unknowns (variables), and your mission is to figure out what those unknowns are. Graphically, each linear equation represents a straight line, and the solution to the system is the point where these lines intersect. That point (x, y) satisfies both equations simultaneously.

Now, there are several methods to solve these systems, but today, we’re focusing on elimination, which is super handy when you can manipulate the equations to cancel out one of the variables. This method shines when coefficients (the numbers in front of the variables) can be easily matched or made opposites. In our case, we have 4x + 2y = -2 and -9x - 5y = 1. The goal of elimination is to manipulate these equations so that either the 'x' coefficients or the 'y' coefficients are the same number but with opposite signs (e.g., 5 and -5). This way, when you add the equations together, one variable will disappear, leaving you with a single equation in one variable, which is much easier to solve. It’s like magic, but it’s actually just math! So, stick with me as we explore how to make this magic happen with our example equations. We'll go step by step to ensure you grasp the concept and can apply it to other similar problems. Trust me, once you get the hang of this, you'll be solving systems of equations with confidence.

Setting Up for Elimination

Alright, let’s get our hands dirty with our example: 4x + 2y = -2 and -9x - 5y = 1. The first step in setting up for elimination is to decide which variable we want to eliminate. Looking at our equations, neither the 'x' coefficients (4 and -9) nor the 'y' coefficients (2 and -5) are easily made the same or opposites with a simple multiplication. So, we'll need to do a bit more work. A strategic approach here is to look for the smallest numbers involved and think about their least common multiple (LCM). This will minimize the size of the numbers we're dealing with, making calculations easier.

For instance, we could eliminate 'y'. The LCM of 2 and 5 is 10. So, we need to transform the 'y' coefficients into 10 and -10. To do this, we can multiply the first equation (4x + 2y = -2) by 5 and the second equation (-9x - 5y = 1) by 2. Remember, what we do to one term in the equation, we must do to all terms to maintain the equality. This is a crucial concept in algebra, ensuring we don’t change the solutions of our system. By multiplying the equations appropriately, we’re setting ourselves up for the next step, where we'll actually eliminate one of the variables. So, get ready to distribute those multiplication factors, and watch as our equations transform into a form that's ready for elimination! This is where the puzzle pieces start falling into place, and you'll see how the method beautifully simplifies the problem.

Multiplying Equations

Okay, guys, we're now at a crucial step: multiplying the equations. As we decided, we're aiming to eliminate 'y'. To achieve this, we're going to multiply the first equation (4x + 2y = -2) by 5 and the second equation (-9x - 5y = 1) by 2. Remember, the goal here is to make the 'y' coefficients opposites, so that when we add the equations together, the 'y' terms will cancel out. This is where precision is key, so let's take it step by step.

First, let’s multiply the entire first equation (4x + 2y = -2) by 5. This means multiplying each term in the equation by 5. So, 5 * (4x) gives us 20x, 5 * (2y) gives us 10y, and 5 * (-2) gives us -10. Our new first equation is now 20x + 10y = -10. See how each term got its fair share of the multiplication? Next, we'll do the same for the second equation. We're multiplying (-9x - 5y = 1) by 2. So, 2 * (-9x) is -18x, 2 * (-5y) is -10y, and 2 * (1) is 2. This gives us the new second equation: -18x - 10y = 2. Notice how the 'y' terms now have coefficients of +10 and -10? That’s exactly what we wanted! By strategically multiplying, we've created a setup where the 'y' terms are ready to be eliminated. The next step is where the magic truly happens, as we add these transformed equations together and watch a variable disappear. So, let's keep this momentum going and move on to the elimination phase!

Eliminating a Variable

Alright, the moment we've been working towards is here: eliminating a variable! We've transformed our original equations into 20x + 10y = -10 and -18x - 10y = 2. Now, watch how beautifully this works. We're going to add these two equations together, combining like terms. This is the heart of the elimination method, where we strategically make one variable vanish, leaving us with a simpler equation.

Let's line them up and add: (20x + 10y) + (-18x - 10y) = -10 + 2. When we combine the 'x' terms, 20x plus -18x gives us 2x. Now, for the 'y' terms, we have 10y plus -10y. These cancel each other out perfectly! This is the elimination in action. On the right side of the equation, -10 plus 2 equals -8. So, our new, simplified equation is 2x = -8. Isn't that satisfying? We've gone from a system of two equations with two variables to a single equation with just one variable. This is a massive simplification, making the problem much easier to solve. By adding the equations strategically, we've bypassed a lot of complexity and zeroed in on a single variable. This step is crucial in solving systems of equations, as it transforms a daunting problem into a manageable one. Now that we have a simple equation, we're just a quick step away from finding the value of 'x'. So, let’s keep the momentum going and solve for 'x'!

Solving for x

We've successfully eliminated 'y' and are left with the equation 2x = -8. Solving for x at this point is straightforward. Our goal is to isolate 'x' on one side of the equation. Currently, 'x' is being multiplied by 2. To undo this multiplication, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 2. Remember, whatever we do to one side of the equation, we must do to the other to maintain balance. This is a fundamental principle in algebra, ensuring we don’t alter the equation’s solution.

So, let's divide both sides of 2x = -8 by 2. On the left side, 2x divided by 2 simplifies to x. On the right side, -8 divided by 2 gives us -4. Therefore, we find that x = -4. That's it! We've solved for 'x'. This is a significant milestone in our journey to solving the system of equations. By performing this simple division, we've unveiled the value of one of our variables. This step highlights the power of algebraic manipulation in simplifying equations and revealing their solutions. Now that we know the value of 'x', the next logical step is to use this information to find the value of 'y'. We're halfway there, guys! Let’s keep pushing forward and find the other piece of our solution.

Solving for y

Now that we've cracked the code for 'x' and found that x = -4, it’s time to hunt down the value of 'y'. Solving for y involves taking the value we found for 'x' and substituting it back into one of our original equations. It doesn't matter which original equation you choose; you'll get the same answer for 'y' either way. However, it’s often a good strategy to pick the equation that looks simpler or has smaller coefficients, as this can reduce the chances of making a calculation error. For our system, the first original equation, 4x + 2y = -2, seems a bit easier to work with.

So, let's substitute x = -4 into the equation 4x + 2y = -2. Replacing 'x' with -4 gives us 4*(-4) + 2y = -2. Now, we simplify: 4 multiplied by -4 is -16, so we have -16 + 2y = -2. Our next goal is to isolate the term with 'y' on one side of the equation. To do this, we'll add 16 to both sides of the equation. This cancels out the -16 on the left side, leaving us with 2y = -2 + 16, which simplifies to 2y = 14. We're almost there! Now, just like when we solved for 'x', we need to isolate 'y'. Currently, 'y' is being multiplied by 2, so we'll divide both sides of the equation by 2. This gives us y = 14 / 2, which simplifies to y = 7. Fantastic! We've found that y = 7. We now have both the 'x' and 'y' values that satisfy our system of equations. We’re in the home stretch, with just one step left to ensure our solution is correct.

Checking the Solution

We've done the hard work of solving for 'x' and 'y', but before we declare victory, it's super important to check our solution. This step is like the quality control of our math – it ensures that our answers are correct and that we haven't made any sneaky mistakes along the way. To check our solution, we'll plug the values we found, x = -4 and y = 7, into both of our original equations. If both equations hold true, then we know we've nailed it!

Let's start with the first equation, 4x + 2y = -2. Substituting x = -4 and y = 7 gives us 4*(-4) + 2*(7) = -2. Simplifying, we get -16 + 14 = -2, which is indeed true. So far, so good! Now, let's test our values in the second equation, -9x - 5y = 1. Substituting x = -4 and y = 7 gives us -9*(-4) - 5*(7) = 1. Simplifying, we get 36 - 35 = 1, which is also true! Since our values for 'x' and 'y' satisfy both original equations, we can confidently say that we've found the correct solution. Our solution is the ordered pair (-4, 7). This check step is not just a formality; it's a powerful tool for catching errors and building confidence in our problem-solving abilities. By taking the time to verify our work, we ensure that we're presenting accurate solutions. And that, my friends, is what makes us math champions!

Conclusion

Woohoo! We've successfully solved the system of linear equations 4x + 2y = -2 and -9x - 5y = 1 using the elimination method. We found that x = -4 and y = 7, which we verified by plugging these values back into our original equations. The key takeaways here are: elimination is a powerful technique for solving systems of equations, and checking your solution is a crucial final step. Remember, guys, solving these problems is like piecing together a puzzle. Each step, from setting up the equations to checking the solution, plays a vital role in the process. With practice, you'll become more comfortable and confident in your ability to tackle these types of problems. So, keep practicing, and don't be afraid to try different approaches. You've got this! Now go out there and conquer those systems of equations!