Solving Systems Of Equations: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever get stuck trying to solve a system of equations? Don't worry, it happens to the best of us! In this article, we're going to break down a specific system of equations and walk through the steps to find the solution. We'll focus on using the method of elimination, which involves creating additive inverses for the coefficients of one of the variables. So, grab your pencils and let's dive in!

The System of Equations

Okay, let's start by taking a look at the system we're going to solve. We have two equations with two variables, x and y:

• Equation 1: $5x - 2y = -11$
• Equation 2: $-2x + 5y = 17$

The goal here is to find the values of x and y that satisfy both equations simultaneously. There are a few different methods we can use to do this, but today we're focusing on elimination. The elimination method works by manipulating the equations so that when we add them together, one of the variables cancels out. This leaves us with a single equation with one variable, which we can easily solve. Once we have the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

Before we jump into the specific steps for this system, let's quickly review the concept of additive inverses. Two numbers are additive inverses if their sum is zero. For example, 5 and -5 are additive inverses because 5 + (-5) = 0. Similarly, -2 and 2 are additive inverses. In the context of solving systems of equations, we want to create additive inverses for the coefficients of either x or y. This will allow us to eliminate one of the variables when we add the equations together. This key concept is the foundation of the elimination method, and understanding it will make the rest of the process much smoother. So, keep in mind that our mission is to find a way to make either the x coefficients or the y coefficients additive inverses of each other. This might involve multiplying one or both equations by a constant. We'll see how this works in the next step!

Step 1: Creating Additive Inverse x-Coefficients

Our first step is to figure out what to multiply Equation 1 by to create additive inverse x-coefficients. Remember, additive inverses are numbers that add up to zero. Currently, the x-coefficient in Equation 1 is 5, and the x-coefficient in Equation 2 is -2. To make these additive inverses, we need to find a number that, when multiplied by 5, will give us the opposite of -2 (which is 2). Alternatively, we could find a number that, when multiplied by -2, gives us the opposite of 5 (which is -5). However, to avoid fractions, it’s often easiest to find the least common multiple (LCM) of the coefficients and then multiply each equation by a factor that will make the coefficients equal to the LCM (with opposite signs).

In this case, the LCM of 5 and 2 is 10. So, we want to transform the x-coefficients into 10 and -10. To do this, we can multiply Equation 1 by 2 and Equation 2 by 5. Multiplying Equation 1 by 2 gives us:

$2 * (5x - 2y) = 2 * (-11)$

Which simplifies to:

$10x - 4y = -22$

Now, let's multiply Equation 2 by 5:

$5 * (-2x + 5y) = 5 * (17)$

Which simplifies to:

$-10x + 25y = 85$

Now, notice what we've accomplished! The x-coefficient in the modified Equation 1 is 10, and the x-coefficient in the modified Equation 2 is -10. These are additive inverses! This is exactly what we wanted. By strategically multiplying each equation by a constant, we've set ourselves up to eliminate the x variable in the next step. Remember, the key to this step is to identify the appropriate multipliers that will create additive inverses for one of the variables. This might require a little trial and error, but with practice, you'll get the hang of it. Now that we have our modified equations with additive inverse x-coefficients, we're ready to move on to the next step: adding the equations together.

Step 2: Adding the Equations

Alright, guys, now comes the super satisfying part – adding the equations together! We've set everything up so that our x terms will cancel out, leaving us with a single equation in terms of y. Let's take a look at our modified equations:

• Modified Equation 1: $10x - 4y = -22$
• Modified Equation 2: $-10x + 25y = 85$

Now, we simply add the left-hand sides of the equations together and the right-hand sides together. This gives us:

$(10x - 4y) + (-10x + 25y) = -22 + 85$

Let's simplify this. The $10x$ and $-10x$ terms cancel each other out (that's the whole point of using additive inverses!), leaving us with:

$-4y + 25y = 63$

Combining the y terms, we get:

$21y = 63$

Boom! We've successfully eliminated x and now we have a simple equation with just y. This is a huge step forward. By adding the equations strategically, we've reduced our system of two equations with two variables to a single equation with one variable. This is a much easier problem to solve. The key takeaway here is the power of elimination. By carefully manipulating the equations, we can eliminate one variable and simplify the problem significantly. Now, all that's left to do in this step is to solve for y. We'll do that in the next part of the article. But before we move on, make sure you understand why adding the equations works and how the additive inverses play a crucial role in eliminating a variable.

Step 3: Solving for y

Okay, we're on the home stretch for finding the value of y! We've got the equation:

$21y = 63$

This is a straightforward equation to solve. To isolate y, we simply need to divide both sides of the equation by 21:

$21y / 21 = 63 / 21$

This simplifies to:

$y = 3$

Fantastic! We've found the value of y. Now we know that y is equal to 3. This is a major accomplishment! We've solved for one of the variables in our system of equations. Remember, our original goal was to find the values of both x and y that satisfy both equations. We're halfway there! The key to this step was recognizing that we had a simple equation with one variable and applying the basic algebraic principle of dividing both sides by the coefficient of the variable. Now that we know the value of y, we can use this information to find the value of x. This is where the substitution part of the process comes in. We'll substitute the value of y into one of the original equations and solve for x. Let's head to the next step to see how it's done!

Step 4: Substituting to Solve for x

Alright, folks, we've found that y = 3. Now it's time to substitute this value back into one of our original equations to solve for x. We can choose either Equation 1 or Equation 2 – it doesn't matter, we'll get the same answer either way. Let's go with Equation 1:

Equation 1: $5x - 2y = -11$

Now, we substitute y = 3 into this equation:

$5x - 2(3) = -11$

Simplifying, we get:

$5x - 6 = -11$

Now, we need to isolate x. First, we add 6 to both sides of the equation:

$5x - 6 + 6 = -11 + 6$

This simplifies to:

$5x = -5$

Finally, we divide both sides by 5:

$5x / 5 = -5 / 5$

Which gives us:

$x = -1$

Yes! We've found the value of x. We now know that x = -1. This is the final piece of the puzzle. We've successfully solved for both x and y. The key to this step was understanding the concept of substitution. Once we knew the value of one variable, we could plug it into an equation to find the value of the other variable. This is a powerful technique that is used in many different areas of mathematics. Now that we have our solution, there's just one thing left to do: check our answer to make sure it's correct.

Step 5: Checking the Solution

Okay, guys, we've got our solution: x = -1 and y = 3. But before we declare victory, it's super important to check our work! This is a crucial step in solving any math problem, especially systems of equations. To check our solution, we need to plug the values of x and y back into both of the original equations. If our solution is correct, it should satisfy both equations. Let's start with Equation 1:

Equation 1: $5x - 2y = -11$

Substitute x = -1 and y = 3:

$5(-1) - 2(3) = -11$

Simplify:

$-5 - 6 = -11$

$-11 = -11$

Great! Our solution works for Equation 1. Now let's check Equation 2:

Equation 2: $-2x + 5y = 17$

Substitute x = -1 and y = 3:

$-2(-1) + 5(3) = 17$

Simplify:

$2 + 15 = 17$

$17 = 17$

Awesome! Our solution also works for Equation 2. Since our solution satisfies both equations, we can confidently say that it is correct. The solution to the system of equations is x = -1 and y = 3. We can write this as an ordered pair: (-1, 3). The importance of checking your solution cannot be overstated. It's a quick and easy way to catch any mistakes you might have made along the way. Think of it as a safety net for your math skills! By plugging your solution back into the original equations, you can be sure that you've found the correct answer.

Conclusion

And there you have it! We've successfully solved the system of equations using the elimination method. We walked through each step, from creating additive inverses to substituting and checking our solution. Remember, solving systems of equations is a fundamental skill in algebra, and mastering it will open doors to more advanced mathematical concepts. So, keep practicing, and don't be afraid to tackle challenging problems. You've got this! Keep checking back with Plastik Magazine for more math tips and tricks. Until next time, happy solving!