Solving Y = 5x - 10 For X = 3.5 A Step-by-Step Guide With Tabular Representation

Hey guys! Today, we're diving into a super fun algebra problem. We're going to solve the equation y = 5x - 10 when x = 3.5. Sounds like a plan? Awesome! But that’s not all, we're also going to represent our findings in a table, making it super clear and easy to understand. So, buckle up and let's get started!

Solving for y when x = 3.5

So, the first thing we need to do is figure out what y is when x is 3.5. Don't worry, it's way easier than it sounds! We're basically just going to swap out the x in our equation with 3.5. Our equation, y = 5x - 10, is going to transform into something super manageable. Are you ready for some substitution magic? Let’s jump right in!

First things first, write down the equation: y = 5x - 10. This is our starting point, our trusty guide in this algebraic adventure. Now, remember that we're trying to find y when x is 3.5. So, every time we see an x, we're going to replace it with 3.5. Think of it like replacing a missing puzzle piece – we're just slotting in the right number to complete the picture. So, let's swap out that x! Our equation now looks like this: y = 5 * (3.5) - 10. See? We've just replaced the x with 3.5, easy peasy!

Now that we've substituted x, it's time for some arithmetic action! We need to follow the order of operations here – you know, the classic PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our case, we've got multiplication and subtraction to deal with. So, what comes first? You guessed it – multiplication! We need to multiply 5 by 3.5. Grab your calculators (or your mental math muscles!) and let’s do this. What’s 5 times 3.5? It’s 17.5! So now our equation looks like this: y = 17.5 - 10. We're getting closer to the finish line, guys!

Alright, we've done the multiplication, now it’s time for the final step: subtraction. We've got 17.5 minus 10 to handle. This is pretty straightforward, right? What's 17.5 take away 10? It’s 7.5! Boom! We've solved for y. So, when x is 3.5, y is 7.5. That wasn't so bad, was it? We've successfully navigated our equation and found the value of y. Give yourselves a pat on the back, mathletes!

Just to recap, we started with the equation y = 5x - 10, we substituted x with 3.5, we did some multiplication (5 * 3.5 = 17.5), and then we finished it off with some subtraction (17.5 - 10 = 7.5). And there you have it: y = 7.5. See how each step logically leads to the next? That’s the beauty of algebra, my friends. So, we've cracked the code for this particular x value. But wait, there's more! We're not stopping here. Next up, we're going to take this solution and represent it in a table. Why? Because visualizing data in a table can make everything even clearer. So, stick around, we're about to make some tabular magic!

Creating a Tabular Representation

Okay, so now that we've solved for y when x is 3.5, it’s time to get visual! We're going to represent our findings in a table. Why a table, you ask? Well, tables are fantastic for organizing data in a clear and concise way. They help us see the relationship between x and y values at a glance. Plus, they're super useful when you want to plot graphs or analyze patterns. So, let’s roll up our sleeves and create a table that showcases our awesome algebra skills!

First things first, let’s talk about what a table looks like. Imagine a grid, kind of like a spreadsheet. We’ll have columns and rows. In our case, we’re going to have two main columns: one for x values and one for y values. These columns will be the headers of our table, telling us what kind of data we're looking at. Think of them as the signposts guiding us through the data landscape. So, at the top of our first column, we'll write “x”, and at the top of the second column, we'll write “y”. Simple as that! We've laid the foundation for our data masterpiece.

Now that we've got our column headers in place, it's time to fill in the data. Remember, we solved for y when x was 3.5. So, that’s the information we’re going to put into our table. Under the x column, we’ll write 3.5. This is the input value, the x that we plugged into our equation. Now, what was the corresponding y value that we calculated? That's right, it was 7.5! So, under the y column, in the same row as 3.5, we’ll write 7.5. See how the table is starting to take shape? We're connecting the dots between x and y values.

Our table might look something like this:

Isn’t that neat? We’ve taken our algebraic solution and transformed it into a visual representation. This table clearly shows us that when x is 3.5, y is 7.5. It's like a little snapshot of our solution, a quick and easy way to see the relationship between x and y. But hey, we don't have to stop at just one point! We can add more rows to our table by solving for y with different x values. This will give us a broader picture of how x and y are related in the equation y = 5x - 10.

Let’s say we wanted to add a couple more x values to our table. For example, what if x was 2? And what if x was 4? We’d just follow the same steps we did before: plug the x value into our equation (y = 5x - 10) and solve for y. Then, we’d add these new x and y values as new rows in our table. Imagine the table growing, expanding to show us more and more of the relationship between x and y. This is how tables can become powerful tools for understanding equations and functions. They let us see the bigger picture, the trends and patterns that emerge as we change the x values.

Expanding the Table with More Values

Alright, guys, let's take our tabular representation to the next level! We've got one data point in our table so far, but let's add a few more to really see how the equation y = 5x - 10 behaves. More data points mean a clearer picture of the relationship between x and y. Think of it like adding more pixels to an image – the more pixels you have, the sharper the image becomes. So, let’s sharpen our understanding by adding some new values to our table!

First, let's pick another value for x. How about x = 2? To find the corresponding y value, we’ll do exactly what we did before: substitute x with 2 in our equation. So, y = 5x - 10 becomes y = 5 * (2) - 10. Now we just need to do the math. What’s 5 times 2? It’s 10! So now we have y = 10 - 10. And what’s 10 minus 10? It’s 0! So, when x is 2, y is 0. We’ve got our next data point. Let's add it to the table!

Our table now looks like this:

See how the table is growing? We're building a collection of x and y pairs that satisfy our equation. Each row represents a solution, a point where the equation holds true. But let’s not stop here! Let’s add another data point to make our picture even clearer. How about we try x = 4? Again, we’ll substitute x with 4 in our equation: y = 5x - 10 becomes y = 5 * (4) - 10. Time for some more arithmetic magic!

What's 5 times 4? It’s 20! So now we have y = 20 - 10. And what’s 20 minus 10? It’s 10! So, when x is 4, y is 10. Awesome! We’ve found another data point to add to our table. Let’s plug it in and see how our table looks now. We're transforming into tabular pros here, guys!

Our updated table looks like this:

Wow! Our table is really starting to take shape. We now have three data points, three x and y pairs that satisfy the equation y = 5x - 10. We can see how y changes as x changes. As x increases, y also increases. This is a clue that we’re dealing with a linear relationship, a straight line when we graph these points. See how tables can help us spot patterns and understand the behavior of equations?

We could keep adding more and more points to our table, but even with just three, we can get a pretty good sense of what’s going on. Tables are super versatile. They’re not just for solving equations; they can be used to represent all sorts of relationships and data. Whether you're tracking sales figures, analyzing scientific data, or even just organizing your grocery list, tables are your friend. They bring order to chaos and make information accessible.

Graphing the Equation from the Table

Okay, guys, we've got a fantastic table filled with x and y values for the equation y = 5x - 10. But guess what? We can take this even further! We're going to use the data in our table to create a graph. Why graph it? Well, graphs are amazing visual tools that can help us see the relationship between x and y in a whole new way. They turn numbers into pictures, making it super easy to spot trends and patterns. So, let’s transform our table into a graph and unlock even more insights!

First things first, let’s talk about the basics of graphing. We’ll need two axes: a horizontal axis called the x-axis and a vertical axis called the y-axis. Think of them as number lines that intersect at a right angle. The x-axis represents our x values, and the y-axis represents our y values. The point where the two axes meet is called the origin, and it’s usually labeled as (0, 0). This is our starting point, our home base in the graphing world. We'll use these axes to plot our data points from the table.

Now, let’s grab the data from our table. Remember, our table looks like this:

Each row in this table gives us a coordinate pair: (x, y). These are the points we're going to plot on our graph. So, we have the points (3.5, 7.5), (2, 0), and (4, 10). Each pair tells us exactly where to place a point on our graph. The first number tells us how far to move along the x-axis, and the second number tells us how far to move along the y-axis. It’s like following a treasure map to find the hidden location!

Let’s start with the first point: (3.5, 7.5). To plot this, we first find 3.5 on the x-axis. Then, we move up vertically until we reach 7.5 on the y-axis. We mark that spot with a dot. That’s it! We’ve plotted our first point. It’s like planting a flag on the graph to mark our territory. Now, let’s move on to the second point: (2, 0). We find 2 on the x-axis, and since the y-value is 0, we don't move up or down at all. We just mark the spot right on the x-axis. Two points down, one to go!

Finally, let’s plot the last point: (4, 10). We find 4 on the x-axis and move up vertically until we reach 10 on the y-axis. We mark that spot with a dot. We’ve done it! We’ve plotted all three points from our table onto our graph. Now, take a step back and look at the arrangement of the points. Do you notice anything? They seem to be lining up in a straight line, don't they? This is a visual confirmation that our equation y = 5x - 10 represents a linear relationship.

If we were to draw a line through these points, it would extend infinitely in both directions. This line is a visual representation of all the possible solutions to our equation. Every point on that line corresponds to an x and y pair that satisfies y = 5x - 10. Isn't that amazing? We've turned an algebraic equation into a geometric shape, a visual masterpiece! Graphing is a powerful tool that can help us understand equations and functions in a more intuitive way.

Conclusion

Alright guys, we've reached the end of our algebraic adventure, and what a journey it's been! We started with the equation y = 5x - 10 and a mission to find the value of y when x is 3.5. We tackled the equation head-on, substituting x with 3.5 and solving for y. We discovered that when x is 3.5, y is 7.5. High five for cracking that code!

But we didn’t stop there! We took our solution and organized it into a table, creating a visual representation of the relationship between x and y. We saw how tables can help us keep track of data and spot patterns. Then, we expanded our table by adding more x values and calculating the corresponding y values. We saw how the table grew, giving us a broader picture of the equation’s behavior. Tables are like our trusty sidekicks in the world of algebra, always there to help us organize and visualize our data.

And then, we went even further! We transformed our table into a graph, turning numbers into a visual masterpiece. We plotted our x and y points on the graph and saw how they lined up in a straight line. This gave us a visual confirmation that our equation represents a linear relationship. Graphing is like turning on the lights in a dark room – it illuminates the relationships between variables and makes everything crystal clear.

So, what have we learned today? We’ve learned how to solve for y in an equation, how to represent our solutions in a table, and how to graph those solutions to visualize the relationship between x and y. We've explored the power of substitution, the clarity of tables, and the visual magic of graphs. These are fundamental skills in algebra, and you guys have nailed them! You're well on your way to becoming algebra wizards!

Remember, math isn't just about numbers and equations; it's about problem-solving, logical thinking, and seeing the connections between things. We've seen how algebra, tables, and graphs are all interconnected, each one building on the others to give us a deeper understanding. So, keep practicing, keep exploring, and keep having fun with math! You've got this!