Solving Systems Of Equations Find The Solution For 2x + 8y = 4 And 6x - Y = -13

Jul 28, 2025 by ADMIN 80 views

Find the Solution of the System of Equations: A Comprehensive Guide

Hey everyone! Today, we're diving into the fascinating world of solving systems of equations. If you've ever felt a bit lost trying to juggle multiple equations at once, don't worry, you're in the right place. We're going to break down a step-by-step approach to tackle these problems with confidence. Specifically, we'll be focusing on solving the following system of equations:

2x + 8y = 4
6x - y = -13

So, buckle up, grab your pencils, and let's get started!

Understanding Systems of Equations

Before we jump into solving, let's take a moment to understand what a system of equations actually is. A system of equations is simply a set of two or more equations that involve the same variables. Our goal is to find values for these variables that satisfy all equations in the system simultaneously. Think of it like finding the sweet spot where all the equations agree.

In our case, we have two equations with two variables, x and y. This is a pretty common scenario, and there are a few different methods we can use to find the solution. We're going to focus on the substitution and elimination methods, as they are powerful and versatile tools in solving systems of equations. Remember, the solution to a system of equations is an ordered pair (x, y) that makes both equations true. It's like finding the coordinates of a point where two lines intersect on a graph. This point represents the unique solution that satisfies both equations.

Why are Systems of Equations Important?

You might be wondering, why should I care about systems of equations? Well, they pop up in all sorts of real-world situations! From calculating costs and profits in business to modeling physical phenomena in science and engineering, systems of equations are essential tools. Imagine you're trying to figure out the right mix of ingredients for a recipe or determining the optimal speed for a train journey. These are the kinds of problems where systems of equations come to the rescue. They allow us to represent complex relationships between different variables and find solutions that meet specific criteria. So, mastering the art of solving systems of equations opens doors to understanding and tackling a wide range of problems in various fields. It's not just about math class; it's about building a powerful problem-solving skill that will serve you well in many aspects of life.

Method 1: The Substitution Method

The substitution method is a fantastic way to solve systems of equations, especially when one of the equations can be easily solved for one variable in terms of the other. The basic idea is to isolate one variable in one equation and then substitute that expression into the other equation. This leaves us with a single equation with a single variable, which we can easily solve. Once we have the value of that variable, we can plug it back into either of the original equations to find the value of the other variable.

Let's apply this method to our system:

2x + 8y = 4
6x - y = -13

Looking at these equations, it seems easier to solve the second equation for y. So, let's do that:

6x - y = -13
-y = -13 - 6x
y = 6x + 13

Now we have y expressed in terms of x. This is the key step in the substitution method. We're going to take this expression for y and substitute it into the first equation:

2x + 8y = 4
2x + 8(6x + 13) = 4

See what we did there? We replaced y in the first equation with the expression 6x + 13. Now we have an equation with only x as the variable. Let's simplify and solve for x:

2x + 48x + 104 = 4
50x + 104 = 4
50x = -100
x = -2

Great! We've found the value of x: x = -2. Now, to find y, we simply plug this value back into either of the original equations or the expression we found for y. Let's use the expression y = 6x + 13:

y = 6(-2) + 13
y = -12 + 13
y = 1

So, we've found that y = 1. Therefore, the solution to the system of equations using the substitution method is (x, y) = (-2, 1).

Tips for Using Substitution Method

The substitution method is a powerful technique, but here are a few tips to keep in mind:

Choose wisely: Look for an equation where one of the variables has a coefficient of 1 or -1. This will make it easier to isolate that variable.
Be careful with signs: When substituting, pay close attention to the signs of the terms. A simple mistake can throw off your entire solution.
Double-check your work: After finding the values of x and y, plug them back into both original equations to make sure they satisfy both.

Method 2: The Elimination Method

The elimination method, also known as the addition method, is another excellent technique for solving systems of equations. The core idea behind this method is to manipulate the equations so that the coefficients of one of the variables are opposites. When we add the equations together, that variable will be eliminated, leaving us with a single equation with a single variable. Sounds pretty neat, right?

Let's revisit our system of equations:

2x + 8y = 4
6x - y = -13

Looking at these equations, we can see that it might be easier to eliminate y. To do this, we need to make the coefficients of y opposites. The coefficient of y in the first equation is 8, and in the second equation, it's -1. To make them opposites, we can multiply the second equation by 8:

8(6x - y) = 8(-13)
48x - 8y = -104

Now we have a new system of equations:

2x + 8y = 4
48x - 8y = -104

Notice that the coefficients of y are now 8 and -8. Perfect! Now we can add the two equations together. When we add the left-hand sides, the y terms will cancel out:

(2x + 8y) + (48x - 8y) = 4 + (-104)
2x + 48x = -100
50x = -100
x = -2

Just like with the substitution method, we've found that x = -2. Now we need to find y. We can plug this value of x back into either of the original equations. Let's use the first equation:

2x + 8y = 4
2(-2) + 8y = 4
-4 + 8y = 4
8y = 8
y = 1

Again, we find that y = 1. So, using the elimination method, the solution to the system of equations is (x, y) = (-2, 1), which is the same result we got with the substitution method!

Key Steps for the Elimination Method

To master the elimination method, remember these key steps:

Choose a variable to eliminate: Look for a variable whose coefficients are either the same or opposites, or can be easily made so by multiplying one or both equations by a constant.
Multiply equations (if needed): Multiply one or both equations by a constant so that the coefficients of the chosen variable are opposites.
Add the equations: Add the equations together. This will eliminate one of the variables.
Solve for the remaining variable: Solve the resulting equation for the remaining variable.
Substitute back: Substitute the value you found back into one of the original equations to solve for the other variable.
Check your solution: Always check your solution by plugging the values of x and y back into both original equations.

Verifying the Solution

It's always a good idea to verify your solution to make sure you haven't made any mistakes along the way. To do this, we simply plug the values we found for x and y back into both of the original equations. If both equations hold true, then we've found the correct solution.

Let's check our solution (x, y) = (-2, 1) with our system of equations:

2x + 8y = 4
6x - y = -13

For the first equation:

2(-2) + 8(1) = 4
-4 + 8 = 4
4 = 4

The first equation holds true! Now let's check the second equation:

6(-2) - (1) = -13
-12 - 1 = -13
-13 = -13

The second equation also holds true! Since our solution satisfies both equations, we can confidently say that (x, y) = (-2, 1) is the correct solution to the system of equations.

The Importance of Verification

Verifying your solution is a crucial step in solving any mathematical problem, not just systems of equations. It's like a safety net that catches any errors you might have made during the process. By plugging your solution back into the original equations, you're essentially testing whether your answer makes sense in the context of the problem. This simple step can save you a lot of time and frustration in the long run. Think of it as the final check before you submit your work or apply your solution to a real-world situation. A little bit of verification can go a long way in ensuring accuracy and confidence in your results. So, always make it a habit to double-check your answers!

Conclusion

Alright, guys, we've covered a lot in this guide! We've explored what systems of equations are, why they're important, and two powerful methods for solving them: the substitution method and the elimination method. We've also emphasized the importance of verifying your solution to ensure accuracy. Remember, solving systems of equations is a valuable skill that can be applied in various fields, so keep practicing and honing your abilities.

To recap, the solution to the system of equations

2x + 8y = 4
6x - y = -13

is (x, y) = (-2, 1). You can use either the substitution or elimination method to arrive at this solution. The key is to understand the underlying principles of each method and choose the one that seems most efficient for the given problem.

So, go ahead and tackle those systems of equations with confidence! You've got the tools and the knowledge to succeed. Keep practicing, and you'll become a pro in no time. And remember, if you ever get stuck, don't hesitate to review the steps and techniques we've discussed. Happy solving!