Solving 3x + Y = 11 And 6x + 2y = 18 With Substitution Method A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of algebra to tackle a common problem: solving a system of linear equations. Specifically, we'll be focusing on the substitution method, a powerful technique that can help you find the values of unknown variables. We'll be working through the equations 3x + y = 11 and 6x + 2y = 18 step-by-step, so you'll not only understand the process but also feel confident applying it to other problems. Let's get started!

Understanding Systems of Linear Equations

Before we jump into the substitution method, let's make sure we're all on the same page about what a system of linear equations actually is. Simply put, it's a set of two or more linear equations that involve the same variables. In our case, we have two equations, and both involve the variables x and y. The goal is to find the values of x and y that satisfy both equations simultaneously. Think of it like finding a secret code that unlocks both equations at the same time. Each linear equation represents a straight line when graphed, and the solution to the system is the point where these lines intersect. If the lines are parallel, there's no solution, and if they're the same line, there are infinitely many solutions. So, systems of linear equations are like a puzzle, and we're going to learn how to solve it using the substitution method. In this particular problem, we have two equations: 3x + y = 11 and 6x + 2y = 18. We need to find the values of x and y that make both of these equations true. Now, why is this important? Well, linear equations pop up everywhere in real life, from calculating costs and distances to modeling complex systems in science and engineering. Mastering how to solve them is a fundamental skill in mathematics and a stepping stone to more advanced concepts. The substitution method is just one way to solve these systems, but it's a really versatile one. It's especially useful when one of the equations can be easily solved for one variable in terms of the other. So, hang in there, because by the end of this guide, you'll be a pro at using substitution to crack these equations! We'll break down each step, explain the logic behind it, and make sure you understand why we're doing what we're doing. That way, you won't just be memorizing steps; you'll be truly understanding the math. Let's get to it!

The Substitution Method: A Step-by-Step Approach

The substitution method is a fantastic tool for solving systems of equations, especially when one equation can be easily rearranged to isolate a variable. It's like a clever detective trick – we isolate one variable and then substitute its value into the other equation, turning a two-variable problem into a single-variable one. Let's break down the general steps and then apply them to our specific problem.

1. Solve one equation for one variable: This is the crucial first step. Look at your equations and identify which one is easiest to manipulate. You want to choose an equation where a variable has a coefficient of 1 or -1, as this will minimize fractions and make the algebra cleaner. In our example, 3x + y = 11, the variable y has a coefficient of 1, making it a prime candidate to isolate. We can rearrange this equation to get y = 11 - 3x. This gives us an expression for y in terms of x. It's like having a secret code for y that we can use in the other equation.
2. Substitute: Now comes the heart of the method. Take the expression you just found and substitute it into the other equation. This is where the magic happens! We're replacing one variable with its equivalent expression, effectively eliminating it from the second equation. In our case, we'll substitute y = 11 - 3x into the equation 6x + 2y = 18. This will give us 6x + 2(11 - 3x) = 18. Notice that we now have an equation with only one variable, x. The problem has transformed from a system of two equations to a single equation that we can solve directly.
3. Solve the resulting equation: With only one variable in the equation, we can now use standard algebraic techniques to find its value. This might involve distributing, combining like terms, and isolating the variable. In our case, we'll distribute the 2 in the equation 6x + 2(11 - 3x) = 18 to get 6x + 22 - 6x = 18. Then, we'll simplify and solve for x. This step is all about using your algebra skills to unravel the equation and reveal the value of x. It's like cracking the code and finding the first piece of the puzzle.
4. Substitute back: Once you've found the value of one variable, it's time to find the other. Take the value you just calculated and substitute it back into either of the original equations (or the rearranged equation from step 1). Choose the equation that looks easiest to work with. This will give you an equation with only the other variable, which you can then solve. This is like using the first piece of the puzzle to find the second. In our example, once we've found x, we'll substitute it back into the equation y = 11 - 3x to find the value of y.
5. Check your solution: Finally, and this is super important, always check your solution! Substitute the values you found for x and y back into both of the original equations. If both equations are true, then you've found the correct solution. If not, you'll need to go back and check your work. This is like making sure your code unlocks both doors – if it doesn't, you know there's a bug somewhere. By following these steps carefully, you can confidently solve any system of equations using the substitution method. It's a powerful and versatile technique that will serve you well in your mathematical journey. Now, let's apply these steps to our specific problem and see how it works in action.

Applying the Substitution Method to 3x + y = 11 and 6x + 2y = 18

Okay, guys, let's get our hands dirty and apply the substitution method to the equations 3x + y = 11 and 6x + 2y = 18. We'll follow the steps we outlined earlier, and I'll walk you through each one to make sure you're crystal clear on what's happening.

Step 1: Solve one equation for one variable

Looking at our equations, 3x + y = 11 seems like the easier one to manipulate. We can easily isolate y by subtracting 3x from both sides. This gives us:

y = 11 - 3x

Awesome! We've now expressed y in terms of x. This is our key to unlocking the solution. We've got a value for y that we can plug into the other equation.

Step 2: Substitute

Now, we take this expression for y (y = 11 - 3x) and substitute it into the second equation, 6x + 2y = 18. This means we replace the y in the second equation with (11 - 3x). So, we get:

6x + 2(11 - 3x) = 18

Notice how we've replaced the y with the expression we found in step 1. This is the crucial substitution step. We've now transformed our problem into a single equation with only one variable, x.

Step 3: Solve the resulting equation

Now, let's solve this equation for x. First, we distribute the 2:

6x + 22 - 6x = 18

Next, we combine like terms. Notice that the 6x and -6x cancel each other out! This leaves us with:

22 = 18

Wait a minute... What's going on here? 22 does not equal 18! This is a bit of a surprise, but it's actually telling us something very important about our system of equations.

Step 4: Analyze the Result

When we arrive at a contradiction like 22 = 18, it means that our system of equations has no solution. Think about it geometrically: this means the two lines represented by our equations are parallel and never intersect. There's no point (x, y) that satisfies both equations at the same time.

Why did this happen?

Let's take a closer look at our original equations:

• 3x + y = 11
• 6x + 2y = 18

If we divide the second equation by 2, we get:

• 3x + y = 9

Now we can see the problem clearly. We have two equations:

• 3x + y = 11
• 3x + y = 9

These equations have the same left-hand side (3x + y), but different right-hand sides (11 and 9). There's no way for the same expression 3x + y to equal both 11 and 9 simultaneously. This is why we ended up with a contradiction.

Key Takeaway:

This example highlights a crucial point: not all systems of equations have a solution. Sometimes, you'll encounter systems that are inconsistent, meaning they have no solution. The substitution method, in this case, helped us identify this inconsistency. So, even though we didn't find specific values for x and y, we still successfully solved the problem by determining that no solution exists. Remember to always be aware of these possibilities when working with systems of equations. Now, while this system had no solution, let's recap the general steps of the substitution method so you're ready for problems that do have solutions!

Recap of the Substitution Method

Alright, guys, even though our specific example led us to a system with no solution, it's super important to solidify the steps of the substitution method in your mind. This method is a powerful tool for solving systems of equations, and you'll definitely use it again and again in your math journey. So, let's do a quick recap to make sure you've got it down.

1. Solve one equation for one variable: Look for the easiest equation to manipulate and isolate one of the variables. Aim for a variable with a coefficient of 1 or -1 to avoid fractions. Get one variable expressed in terms of the other (e.g., y = ... or x = ...).
2. Substitute: Take the expression you just found and substitute it into the other equation. This replaces one variable in the second equation, leaving you with an equation with only one variable.
3. Solve the resulting equation: Use your algebraic skills to solve the single-variable equation. This might involve distributing, combining like terms, and isolating the variable.
4. Substitute back: Once you've found the value of one variable, substitute it back into either of the original equations (or the rearranged equation from step 1) to find the value of the other variable.
5. Check your solution: Always, always, always check your solution! Substitute the values you found for x and y back into both of the original equations. If both equations are true, you've got the correct solution. If not, go back and check your work.

Key takeaways about the substitution method:

• It's most effective when one equation can be easily solved for one variable.
• It transforms a two-variable problem into a single-variable problem.
• It can reveal inconsistencies in the system (like we saw in our example).
• Checking your solution is crucial to ensure accuracy.

By mastering these steps, you'll be well-equipped to tackle a wide range of systems of equations. Practice makes perfect, so try applying the substitution method to different problems and see how it works. Remember, even if you encounter a system with no solution, the method itself is still valuable in identifying that fact. Keep up the great work, and you'll be a substitution pro in no time!

Conclusion

So, guys, we've journeyed through the substitution method, tackling the system 3x + y = 11 and 6x + 2y = 18. While we discovered that this particular system has no solution, the process highlighted the power and versatility of the substitution method. We learned how to isolate a variable, substitute its expression into another equation, and ultimately, determine whether a solution exists. Remember, the substitution method is a valuable tool in your mathematical arsenal, and mastering it will empower you to solve a wide range of problems. Keep practicing, stay curious, and you'll continue to grow your problem-solving skills. You've got this!