Solve Equations: Substitution & Check Solutions

Hey guys! Let's dive into the world of solving systems of equations, specifically using the substitution method. We'll walk through a problem together, step-by-step, making sure we understand how it works and, most importantly, how to check our solutions. This is like a mathematical adventure, and I'm stoked you're here to learn! So, let's get started, shall we?

We are going to solve the following equations by the substitution method and check our solution:

• 0. 5x + 0.25y = 36
• y + 18 = 16x

Understanding the Substitution Method: The Basics

Alright, before we jump into the problem, let's quickly recap what substitution is all about. The substitution method is a fantastic way to solve a system of equations. Essentially, the goal is to solve for one variable in one of the equations and then plug that value into the other equation. It's like a clever little trick that allows us to simplify things and find the values that make both equations true at the same time. Think of it as detective work, where you're trying to uncover the secrets of the equations. We start by isolating one variable in one equation, then use that expression in the other equation. This helps us to get an equation with just one variable, which is easy to solve. Once we find the value of that variable, we plug it back in to find the value of the other variable.

The cool thing about substitution is that it works whether the equations are linear or more complex. It's a versatile tool! It's super important to stay organized and keep track of your steps when you're using this method. Double-check your calculations, especially when dealing with fractions or decimals. This will save you from making silly mistakes. Also, take your time! Don’t rush the process; understanding each step is more important than speed.

So, as we tackle our problem, keep in mind these principles. We want to find values for 'x' and 'y' that make both equations true. It’s like finding a treasure where 'x' and 'y' are the coordinates. So, ready to use this method to solve some equations? Let’s do it!

Step-by-Step: Solving the Equations

Now, let's get our hands dirty and actually solve this system of equations. Here's how we'll do it step-by-step to solve the system by substitution. Trust me, once you get the hang of it, it’s not too bad. The more you practice, the easier it gets!

First, let's write out our equations:

• Equation 1: 0.5x + 0.25y = 36
• Equation 2: y + 18 = 16x

Our first move is to isolate one variable in one of the equations. Looking at Equation 2 (y + 18 = 16x), it looks easier to isolate 'y'. We can rearrange this to solve for 'y'.

y + 18 = 16x y = 16x - 18

Now we've got an expression for 'y' in terms of 'x'. The next step is to substitute this expression into Equation 1. So, wherever we see 'y' in Equation 1, we're going to replace it with '16x - 18'.

Equation 1: 0.5x + 0.25y = 36 Substitute 'y = 16x - 18': 0. 5x + 0.25(16x - 18) = 36

Next, we need to simplify and solve for 'x'. Let's distribute the 0.25:

0. 5x + (0.25 * 16x) - (0.25 * 18) = 36
1. 5x + 4x - 4.5 = 36

Combine like terms:

4. 5x - 4.5 = 36

Add 4.5 to both sides:

5. 5x = 40.5

Finally, divide both sides by 4.5 to isolate 'x':

x = 40.5 / 4.5 x = 9

Woohoo! We've found the value of 'x'. Now, let's use that value to find the value of 'y'.

Finding the Value of Y

We know that x = 9. So, let’s go back to our earlier expression where we isolated y: y = 16x - 18

Now substitute x = 9: y = 16(9) - 18 y = 144 - 18 y = 126

Awesome! We've found that y = 126. So now, we have values for both 'x' and 'y' that we believe satisfy both equations. The solution is the point (9, 126).

Checking Your Solution: The Verification Step

Okay, we’ve got our solution (9, 126), but we’re not done yet! We always need to check our solution to make sure it’s correct. This is a crucial step. It's like double-checking your math to ensure you didn’t make any mistakes. We'll do this by plugging our x and y values back into the original equations. If both equations are true after we plug in the values, we know our solution is correct. If not, it means we made a mistake somewhere along the way, and we need to go back and find it. This step is super important for building confidence in your answers and making sure you really understand what you’re doing.

Let’s start by plugging the values into Equation 1: 0. 5x + 0.25y = 36 Substitute x = 9 and y = 126: 0. 5(9) + 0.25(126) = 36 4. 5 + 31.5 = 36 36 = 36

Equation 1 checks out! Now, let’s check Equation 2: y + 18 = 16x Substitute x = 9 and y = 126: 126 + 18 = 16(9) 144 = 144

Equation 2 checks out too! Both equations are true with x = 9 and y = 126. This means our solution is indeed correct. Great job!

Conclusion: You've Nailed It!

Congratulations, guys! You've successfully solved a system of equations using the substitution method and checked your solution. Remember, the key is to be organized, take your time, and double-check your work. Practice makes perfect, so keep practicing, and you'll become a pro at solving these types of problems. Now that you've got this down, you can tackle more complex equations with confidence. Keep up the awesome work!

So, what did we learn today?

• How to use the substitution method to solve systems of equations.
• How to isolate one variable.
• How to substitute the value of an equation.
• How to check the solution.

Keep practicing, and you'll become a math whiz in no time. If you have any questions, feel free to ask! Happy solving!