Solving For Y: 3x + 8y = 48 When X = 0
Hey guys! Let's dive into a fun math problem today. We're going to tackle an equation and find a missing value. It's like a mini-detective game with numbers! Our main goal is to figure out the value of y when x is zero in the equation 3x + 8y = 48. This might sound a bit intimidating, but trust me, it's easier than you think. We’ll break it down step by step, so you can follow along and even try similar problems on your own. So, grab your thinking caps, and let’s get started!
Understanding the Equation
Before we jump into solving, let’s make sure we understand what the equation 3x + 8y = 48 is telling us. In simple terms, this equation shows a relationship between two variables, x and y. Think of x and y as placeholders for numbers. The equation says that if you multiply the number you put in place of x by 3, and then add that to 8 times the number you put in place of y, the result should be 48.
Why is this important? Well, equations like this one are used everywhere in real life – from calculating the cost of ingredients for a recipe to figuring out how long it will take to travel a certain distance. Understanding how to work with these equations opens up a whole world of problem-solving possibilities.
In our specific problem, we're given a table with a missing y value when x is 0. This is a classic example of a problem where we can use the equation to find the missing piece. By substituting the known value of x into the equation, we can then solve for y. It’s like having a puzzle where we have most of the pieces, and we just need to find the one that fits perfectly. Let's see how we can do this!
Breaking Down the Components
To truly grasp the equation, let’s break it down into its individual parts:
- 3x: This means “3 times x.” It's a shorthand way of writing 3 * x. The number 3 is called a coefficient, and it's multiplied by the variable x.
- 8y: Similarly, this means “8 times y.” The number 8 is the coefficient, and it's multiplied by the variable y.
- = 48: This is the heart of the equation. It tells us that whatever we get when we calculate 3x + 8y, it must be equal to 48. This equality is what allows us to solve for the missing value. Think of it as a balancing scale – the left side (3x + 8y) must weigh the same as the right side (48).
Understanding these components is crucial because it allows us to manipulate the equation and isolate the variable we want to find (y in this case). It’s like learning the individual notes in a musical piece before putting them together to play the melody. Once we understand the individual parts, we can see how they fit together to form the whole equation.
Substituting x = 0 into the Equation
Now comes the exciting part: substituting the value of x into our equation! We know that x = 0, and our equation is 3x + 8y = 48. So, what happens when we replace x with 0?
Let's do it:
- Start with the equation: 3x + 8y = 48
- Replace x with 0: 3(0) + 8y = 48
See how we simply swapped the x with the number 0? This is the core of substitution. We're taking a known value and plugging it into the equation. Now, let's simplify this a bit further.
Simplifying the Equation
We've substituted x = 0 into the equation, and now we have 3(0) + 8y = 48. The next step is to simplify this expression. Remember the order of operations (PEMDAS/BODMAS)? Multiplication comes before addition. So, we need to deal with 3(0) first.
What is 3 times 0? It's 0! Anything multiplied by zero is zero. So, our equation becomes:
0 + 8y = 48
Now, we're getting somewhere! We've eliminated one term from the left side of the equation, making it much simpler. Adding 0 to anything doesn't change its value, so we can rewrite the equation as:
8y = 48
This simplified equation is much easier to work with. We've gone from a slightly complex equation to a simple one-step equation. Now, we just need to isolate y to find its value.
Solving for y
We've arrived at the equation 8y = 48. Our goal now is to isolate y on one side of the equation. This means we want to get y by itself. How do we do that? We use the concept of inverse operations.
What's the inverse operation of multiplication? It's division! Since 8 is being multiplied by y, we need to divide both sides of the equation by 8 to undo the multiplication.
Let's do it:
- Start with the equation: 8y = 48
- Divide both sides by 8: (8y) / 8 = 48 / 8
On the left side, the 8s cancel each other out, leaving us with just y. On the right side, 48 divided by 8 is 6. So, our equation simplifies to:
y = 6
We've done it! We've successfully solved for y. When x is 0, y is 6. It’s like we’ve cracked the code and found the missing piece of our puzzle.
Checking Our Solution
It's always a good idea to check our solution to make sure it's correct. This is like double-checking your work on a test. To check our solution, we'll plug x = 0 and y = 6 back into the original equation and see if it holds true.
Original equation: 3x + 8y = 48
Substitute x = 0 and y = 6: 3(0) + 8(6) = 48
Simplify:
- 3(0) = 0
- 8(6) = 48
- 0 + 48 = 48
So, we have 48 = 48, which is true! This confirms that our solution is correct. We can confidently say that when x is 0, y is indeed 6.
The Answer and Why It Matters
So, the missing value in the table is 6! This corresponds to option A in the multiple-choice options provided. We've successfully solved the problem by understanding the equation, substituting the given value, simplifying, and isolating the variable we wanted to find.
But why does this matter? Well, this type of problem is a foundational concept in algebra. It helps us understand how variables relate to each other and how we can manipulate equations to solve for unknowns. These skills are essential for more advanced math topics and have real-world applications in fields like science, engineering, economics, and even everyday problem-solving. Think about calculating your budget, figuring out the best deal on a purchase, or even planning a road trip – these all involve algebraic thinking!
Real-World Applications
Let's think about a real-world example. Imagine you're planning a party and you have a budget of $48. You want to buy some pizzas that cost $3 each and some drinks that cost $8 per pack. The equation 3x + 8y = 48 could represent this situation, where x is the number of pizzas and y is the number of packs of drinks. If you decide you don't want to buy any pizzas (x = 0), how many packs of drinks can you buy? We just solved that – you can buy 6 packs of drinks!
This is just one simple example, but it illustrates how algebraic equations can be used to model real-world scenarios and solve practical problems. By understanding the concepts we've discussed today, you're building a strong foundation for tackling more complex problems in the future.
Practice Makes Perfect
Now that we've walked through this problem together, the best way to solidify your understanding is to practice! Try solving similar equations with different values for x and y. You can even create your own equations and challenge yourself or your friends.
Here are a few ideas to get you started:
- Solve for x when y = 0 in the equation 3x + 8y = 48.
- Try a different equation, like 2x + 5y = 20, and solve for y when x = 5.
- Think about other real-world scenarios where you could use an equation like this, and try to set up the equation and solve for the unknowns.
The more you practice, the more comfortable you'll become with these concepts. And remember, math is like learning a new language – it takes time and effort, but it's totally worth it in the end!
Final Thoughts
So, guys, we've successfully navigated this equation and found the missing value. We started by understanding the equation, then substituted the given value of x, simplified the equation, and finally solved for y. We even checked our answer to make sure it was correct! This is a great example of how we can use algebra to solve problems and find missing information. Keep practicing, keep exploring, and remember that math can be fun! You've got this!