Raffle Tickets: Solve, Tabulate, And Plot 2x + 1.5y = 18
Hey guys! Let's dive into a fun math problem involving raffle tickets! Imagine you're at a club event, and there are two types of raffle tickets you can buy: book raffle tickets and food raffle tickets. The equation 2x + 1.5y = 18 represents the relationship between the number of book raffle tickets (x) and food raffle tickets (y) you can purchase. Our mission today is to solve this equation for y, create a table of ordered pairs, and then plot those pairs on a graph. Buckle up, because we're about to embark on a mathematical adventure!
Solving the Equation for y
Our first task is to isolate y in the equation 2x + 1.5y = 18. This will allow us to easily determine the number of food raffle tickets we can buy for any given number of book raffle tickets. Think of it like rearranging the furniture in a room to make it more functional. We're just rearranging the equation to make it more useful for our purposes. Here’s how we do it, step by step:
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Subtract 2x from both sides: To get the term with y by itself, we need to get rid of the 2x term on the left side. We do this by subtracting 2x from both sides of the equation. This maintains the balance of the equation, like making sure both sides of a scale weigh the same. So, we have:
1. 5y = 18 - 2x
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Divide both sides by 1.5: Now, we want to isolate y completely. It’s currently being multiplied by 1.5, so to undo that, we divide both sides of the equation by 1.5. This is like cutting a pizza into equal slices; we're dividing both sides equally. This gives us:
y = (18 - 2x) / 1.5
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Simplify: To make the equation even cleaner, we can simplify the right side. Divide 18 by 1.5, which equals 12, and divide -2 by 1.5, which equals -4/3. Therefore, the simplified equation is:
y = 12 - (4/3)x
Now we have our equation solved for y! This equation, y = 12 - (4/3)x, is super helpful because it tells us exactly how many food raffle tickets (y) we can buy if we know how many book raffle tickets (x) we want. It’s like having a secret code that unlocks the possibilities for our raffle ticket purchases.
Creating an Input-Output Table
Next up, let's create an input-output table. This table will help us organize different values of x (the input, or number of book raffle tickets) and their corresponding values of y (the output, or number of food raffle tickets). Think of it as a menu where you choose the number of book tickets and the table tells you how many food tickets you can get. Choosing smart values for x will make the calculations easier, especially since we have a fraction in our equation. We want values that, when multiplied by 4/3, result in whole numbers. Here are a few values we can use:
- x = 0: If we buy zero book raffle tickets, this will help us find the maximum number of food raffle tickets we can purchase.
- x = 3: Choosing 3 makes the fraction (4/3) nicely simplify, making our calculations easier.
- x = 6: Similar to 3, this value will also make the fraction calculation straightforward.
- x = 9: This is another multiple of 3, ensuring we get a whole number result for y.
Now, let's plug these values of x into our equation y = 12 - (4/3)x and calculate the corresponding values of y:
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If x = 0:
y = 12 - (4/3)(0) = 12 - 0 = 12
So, if we buy 0 book raffle tickets, we can buy 12 food raffle tickets.
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If x = 3:
y = 12 - (4/3)(3) = 12 - 4 = 8
If we buy 3 book raffle tickets, we can buy 8 food raffle tickets.
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If x = 6:
y = 12 - (4/3)(6) = 12 - 8 = 4
If we buy 6 book raffle tickets, we can buy 4 food raffle tickets.
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If x = 9:
y = 12 - (4/3)(9) = 12 - 12 = 0
If we buy 9 book raffle tickets, we can buy 0 food raffle tickets.
Let's summarize these findings in a table:
| Input (x) | Output (y) |
|---|---|
| 0 | 12 |
| 3 | 8 |
| 6 | 4 |
| 9 | 0 |
This table is like our personal raffle ticket guide! It clearly shows us the relationship between the number of book raffle tickets and food raffle tickets we can buy. Isn't math cool?
Plotting the Ordered Pairs
Alright, we're on the final stretch! Now we're going to take the ordered pairs we found in our table and plot them on a graph. Think of the graph as a visual representation of our raffle ticket options. It's like a map showing us the different combinations of book and food tickets we can purchase.
To plot these points, we'll use a coordinate plane. The x-axis (horizontal axis) will represent the number of book raffle tickets, and the y-axis (vertical axis) will represent the number of food raffle tickets. Each ordered pair (x, y) corresponds to a point on this plane.
Let's plot the points from our table:
- (0, 12): Start at the origin (0,0). Since x is 0, we don't move horizontally. Since y is 12, we move 12 units up the y-axis. Mark this point.
- (3, 8): Start at the origin. Move 3 units to the right along the x-axis (because x is 3) and then 8 units up the y-axis (because y is 8). Mark this point.
- (6, 4): Start at the origin. Move 6 units to the right along the x-axis and then 4 units up the y-axis. Mark this point.
- (9, 0): Start at the origin. Move 9 units to the right along the x-axis. Since y is 0, we don't move vertically. Mark this point on the x-axis.
Once we've plotted all the points, we'll notice that they all lie on a straight line. This is because our equation 2x + 1.5y = 18 is a linear equation. To complete our graph, we can draw a line through these points. This line represents all the possible combinations of book and food raffle tickets we can buy with our budget.
This graph gives us a clear visual understanding of our options. We can see that as we buy more book raffle tickets, we can buy fewer food raffle tickets, and vice versa. It's a fantastic way to see the relationship between the two types of tickets.
Conclusion
Awesome job, guys! We've successfully solved the equation 2x + 1.5y = 18 for y, created an input-output table, and plotted the ordered pairs on a graph. We've transformed a simple equation into a comprehensive understanding of our raffle ticket options. We learned how to rearrange equations, organize data in a table, and visualize data on a graph. Math can be super fun and incredibly useful in everyday situations, like figuring out the best way to spend our money on raffle tickets! Keep exploring the world of math, and you'll be amazed at what you can discover.
