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

Hey math enthusiasts! Today, we're diving into a fundamental concept in algebra: solving systems of equations using the substitution method. Understanding this method is crucial, as it unlocks the ability to tackle various mathematical problems. Let's break down the process and solve the system: $3x - 8y = -1$ and $x = 5$. Don't worry, it's not as scary as it sounds! I'll guide you through each step, making it easy to grasp. We'll be using substitution to find the values of x and y that satisfy both equations simultaneously. So, grab your pencils and let's get started!

Understanding Systems of Equations and Substitution

Alright guys, let's start with the basics. A system of equations is simply a set of two or more equations that we aim to solve together. The solution to a system is the set of values for the variables (in this case, x and y) that make all the equations true. There are several methods for solving systems of equations, but today we're focusing on substitution. The substitution method is a powerful technique where we solve one equation for one variable and then substitute that expression into the other equation. This process eliminates one of the variables, allowing us to solve for the remaining variable. Once we find the value of one variable, we can substitute it back into either of the original equations to solve for the other variable. Think of it like a puzzle – we use information from one piece to solve for another, eventually completing the whole picture.

So, why use substitution? Well, it's particularly useful when one of the equations is already solved for a variable, or when it's easy to isolate a variable in one of the equations. In our example, the second equation, $x = 5$, gives us the value of x directly. This makes substitution a straightforward and efficient approach. Remember, the goal is always to find the values of the variables that satisfy all the equations in the system. The substitution method provides us with a systematic way to achieve this. It's like having a secret weapon in your algebra arsenal! It might seem complex at first, but with practice, it becomes second nature. And trust me, it's super satisfying when you finally find the solution.

Step-by-Step Solution: Let's Do This!

Okay, let's get down to business and solve our system of equations using the substitution method. Remember our equations: $3x - 8y = -1$ and $x = 5$. Here’s a breakdown of the steps:

1. Substitution: Since we already know that $x = 5$, we'll substitute this value into the first equation: $3x - 8y = -1$. Replacing x with 5, we get: $3(5) - 8y = -1$. See? Easy peasy!

2. Simplify: Now, let's simplify the equation. Multiply 3 by 5: $15 - 8y = -1$. We're getting closer to isolating y.

3. Isolate the variable: Our next goal is to isolate y. Subtract 15 from both sides of the equation: $15 - 8y - 15 = -1 - 15$. This simplifies to: $-8y = -16$.

4. Solve for y: To find the value of y, divide both sides of the equation by -8: $-8y / -8 = -16 / -8$. This gives us: $y = 2$.

5. Solution: Now that we have the values of both x and y, we can express the solution as an ordered pair (x, y). In our case, the solution is (5, 2). This means that when x is 5 and y is 2, both equations in the system are true. Congrats, we solved it!

To make sure you understand it, go through the above steps again, slowly. Practice is key, and the more you practice, the better you'll become at solving these types of problems. Remember, the main idea is to use one equation to eliminate a variable in the other. Once you get the hang of it, you'll find that solving systems of equations is not only manageable but also a lot of fun!

Verifying the Solution: Double-Checking Our Work

Always a good idea, right? Now that we've found our solution, let's double-check to make sure we're spot on. Verifying the solution is an important step to ensure our answer is correct. We do this by plugging the values of x and y back into the original equations and checking if both equations hold true. Let's do it!

First, let's plug the values into the first equation: $3x - 8y = -1$. We know $x = 5$ and $y = 2$, so we substitute those values: $3(5) - 8(2) = -1$. Simplifying this gives us: $15 - 16 = -1$. And, lo and behold, this simplifies to: $-1 = -1$. The first equation checks out! Everything looks good!

Now, let's check the second equation, which is $x = 5$. Since we found that $x = 5$, this equation is already confirmed to be true. Both equations are satisfied with the values x = 5 and y = 2. Great job, guys! This confirms that our solution (5, 2) is indeed correct. Always take this extra step – it's a great way to catch any potential errors and build your confidence in your problem-solving skills. So, whenever you solve a system of equations, don't forget to plug your solution back into the original equations to make sure everything lines up. It's like the final stamp of approval on your hard work.

Different Scenarios and Things to Keep in Mind

Alright, let's talk about some different scenarios you might encounter and some tips to help you along the way. While the process of substitution remains the same, the nature of the equations might vary. Understanding these variations will make you a more versatile problem solver. Not all systems of equations have a unique solution. Some systems might have no solution, while others might have infinitely many solutions. If you encounter a situation where the variables cancel out and you end up with a false statement (like 2 = 5), the system has no solution. This means there are no values of x and y that can satisfy both equations. If, on the other hand, the variables cancel out and you end up with a true statement (like 0 = 0), the system has infinitely many solutions. This means the two equations represent the same line.

Also, you might not always be given an equation that's already solved for a variable. In such cases, you’ll need to solve one of the equations for either x or y first, before substituting. Choose the equation and variable that seem easiest to isolate. This can often involve a bit of algebraic manipulation, but the goal remains the same: to get one variable by itself on one side of the equation. So, keep an eye out for opportunities to simplify your work. Sometimes, a little bit of rearranging can make the whole problem much easier.

Finally, be careful with your signs and coefficients. A small mistake in arithmetic can lead to the wrong answer. Double-check your calculations, especially when dealing with negative numbers. If you're feeling a bit rusty with your algebra skills, don't worry. Practice makes perfect! Working through various examples and problem sets can help you master these concepts. Remember, the key is to stay organized, pay attention to detail, and take your time.

Conclusion: You've Got This!

So there you have it, guys! We've successfully navigated the world of solving systems of equations using the substitution method. I hope you found this guide helpful and easy to follow. Remember, the key is to substitute one equation into the other to eliminate a variable and solve for the remaining one. Once you find the value of one variable, plug it back into either of the original equations to solve for the other. Always verify your solution by plugging the values back into the original equations to ensure everything is correct.

This method is a foundational concept in algebra and has applications in various areas of mathematics and science. As you practice more and more, you'll become more confident in your ability to solve these types of problems. Keep practicing and exploring different types of systems and equations. The more you work with these concepts, the better you'll understand them. Math can be challenging, but it's also incredibly rewarding. Keep up the amazing work, and don't be afraid to ask questions. You've got this!

Keep exploring and happy solving! You're now equipped with a powerful tool for solving systems of equations. Go forth and conquer those equations! Don't forget to practice, practice, practice. You'll become a pro in no time! Remember, math is like any other skill – the more you practice, the better you get. So, keep at it, and you'll see amazing results. Best of luck on your mathematical journey, and I hope this guide helps you every step of the way!