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

Let's dive into the fascinating world of mathematics and tackle a system of linear equations. Today, we're going to unravel the mystery behind the equations:

-2x - 3y = -28 -2x + 2y = 22

Our goal is to find the specific values for 'x' and 'y' that satisfy both equations simultaneously. This is like finding the single point where two lines intersect on a graph. There are a few common methods to solve such systems, including substitution and elimination. We'll be using the elimination method today, as it seems particularly well-suited for these equations because the 'x' terms have the same coefficient.

Understanding the Elimination Method

The elimination method is a powerful technique used to solve systems of equations by strategically adding or subtracting the equations to eliminate one of the variables. The core idea is to manipulate the equations so that when you combine them, one variable cancels out, leaving you with a simpler equation that can be solved for the remaining variable. Once you have the value of one variable, you can substitute it back into either of the original equations to find the value of the other. It's a systematic approach that, with a little practice, becomes quite intuitive. We'll walk through each step with our given equations, making sure to explain the reasoning behind every move. This method is especially useful when the coefficients of one variable are the same or can easily be made the same through multiplication. Let's get started on finding that solution!

Step 1: Analyze and Prepare the Equations

When we look at our system:

Equation 1: -2x - 3y = -28 Equation 2: -2x + 2y = 22

We notice that both equations have a -2x term. This is fantastic because it means we can eliminate 'x' by simply subtracting one equation from the other. If the coefficients weren't the same, our first step would be to multiply one or both equations by a number to make them match. For instance, if we had 2x in one and 3x in the other, we might multiply the first equation by 3 and the second by 2 to get 6x in both. But here, we're lucky! The -2x terms are already identical, setting us up perfectly for elimination. It's always a good idea to write the equations aligned vertically, with the x-terms, y-terms, and constants lined up. This visual arrangement makes it much easier to see how the elimination will work. Think of it like preparing your ingredients before you start cooking; having everything in order makes the process smooth and less prone to errors. So, with our equations neatly aligned, we're ready to perform the subtraction that will begin to unravel our solution.

Step 2: Eliminate One Variable

Now, let's perform the elimination. Since both equations have -2x, we can subtract Equation 2 from Equation 1. This is how it looks:

(-2x - 3y) - (-2x + 2y) = -28 - 22

Let's distribute the negative sign in the first part:

-2x - 3y + 2x - 2y = -28 - 22

Notice how the -2x and +2x terms cancel each other out, leaving us with:

-5y = -50

This is the magic of elimination! We've successfully removed 'x' from the equation, leaving us with a much simpler equation containing only 'y'. This step is crucial because it simplifies the problem dramatically. If we had chosen to add the equations, we would have ended up with -4x - y = -6, which doesn't eliminate a variable immediately. Subtracting was the right move here. The goal is always to make one variable disappear so you can isolate the other. It’s like solving a puzzle where you remove pieces that don’t fit to reveal the correct picture. The resulting equation, -5y = -50, is straightforward to solve.

Step 3: Solve for the Remaining Variable

With the equation -5y = -50, we can now easily solve for 'y'. To isolate 'y', we need to divide both sides of the equation by -5:

y = -50 / -5

y = 10

And there we have it! We've found the value of 'y'. It's 10. This is a significant breakthrough. We've successfully determined one half of our solution. Remember, the aim is to find both 'x' and 'y', and getting one value is a huge step forward. This equation was simple enough that division did the trick. If we had ended up with something like 3y = 15, the process would be the same: divide by 3. The key is to perform the inverse operation to isolate the variable. In this case, since 'y' was multiplied by -5, we divide by -5. This gives us the value of 'y' directly. It's important to be careful with signs during this step; a common mistake is to misplace a negative sign, which would lead to an incorrect answer. But here, -50 divided by -5 correctly yields a positive 10.

Step 4: Substitute Back to Find the Other Variable

Now that we know y = 10, we can substitute this value back into either of the original equations to find 'x'. Let's use Equation 2, as it has positive coefficients for 'y', which might make substitution slightly easier:

-2x + 2y = 22

Substitute y = 10:

-2x + 2(10) = 22

-2x + 20 = 22

Now, we need to isolate '-2x'. Subtract 20 from both sides:

-2x = 22 - 20

-2x = 2

Finally, to solve for 'x', divide both sides by -2:

x = 2 / -2

x = -1

So, the value of 'x' is -1. We've now found both variables! This substitution step is where you bring your two equations back together to find the complete solution. Choosing either original equation should yield the same result for 'x'. If you get a different value for 'x' using Equation 1, it's a strong indicator that there might have been a calculation error in Step 2, Step 3, or during this substitution. It's always a good practice to check your work by substituting the values into the other equation as well, just to be absolutely sure.

Step 5: Verify the Solution

To be completely certain that our solution x = -1 and y = 10 is correct, we should substitute these values back into both of the original equations. This is the final check, ensuring that our hard work has paid off.

Let's check Equation 1: -2x - 3y = -28 Substitute x = -1 and y = 10:

-2(-1) - 3(10) = -28

2 - 30 = -28

-28 = -28

This equation holds true! Now let's check Equation 2: -2x + 2y = 22 Substitute x = -1 and y = 10:

-2(-1) + 2(10) = 22

2 + 20 = 22

22 = 22

Both equations are satisfied! This means our solution is absolutely correct. The solution to the system of equations is x = -1 and y = 10. This verification step is essential in mathematics. It's not just about finding an answer; it's about proving that the answer is correct. Think of it as a double-check in any important task. You wouldn't want to submit a report without proofreading it, right? Similarly, in math, verification is your proof that you've got it right. It reinforces understanding and builds confidence in your problem-solving abilities. This systematic approach, from analysis to verification, is the hallmark of good mathematical practice.

Conclusion

We've successfully navigated the process of solving a system of linear equations using the elimination method. By carefully analyzing the equations, eliminating one variable, solving for the other, and then substituting back, we arrived at the unique solution: x = -1 and y = 10. Remember, the key to mastering these problems lies in understanding each step and practicing consistently. Don't be discouraged if you make mistakes along the way; they are valuable learning opportunities. The ability to solve systems of equations is a fundamental skill in algebra, opening doors to understanding more complex mathematical concepts and real-world applications, from economics to engineering. Keep practicing, and you'll find these problems become second nature!

For further exploration into solving systems of equations and other algebraic concepts, you might find the resources at Khan Academy very helpful. They offer a vast array of free lessons and practice exercises covering many topics in mathematics.