Completing Ordered Pairs For The Linear Equation Y = (1/5)x A Step-by-Step Guide

Hey guys! Today, we're going to dive into a fun and fundamental concept in mathematics: linear equations and how to complete tables of ordered pairs. Specifically, we'll be working with the equation y = (1/5)x. This equation represents a straight line when graphed, and our goal is to find the missing y values for given x values, and vice versa, to complete the ordered pairs in a table. This skill is super important because it helps us understand the relationship between x and y in a linear equation, and it's a stepping stone to graphing lines and solving more complex problems. So, let's jump right in and make sure we nail this concept!

Understanding Linear Equations and Ordered Pairs

Before we fill in the table, let's quickly recap what linear equations and ordered pairs are all about. A linear equation, like our y = (1/5)x, is an equation where the highest power of the variables (in this case, x and y) is 1. This means when we graph the equation, we'll get a straight line. Now, an ordered pair is simply a pair of numbers, written as (x, y), that represents a point on that line. The x value tells us how far to move horizontally from the origin (0, 0), and the y value tells us how far to move vertically. Each point on the line satisfies the equation, meaning if we plug the x and y values of the point into the equation, it will hold true. For example, if we have the point (5, 1), plugging these values into our equation gives us 1 = (1/5) * 5, which simplifies to 1 = 1, a true statement. This means (5, 1) is indeed a point on the line represented by y = (1/5)x. Understanding this relationship between linear equations, ordered pairs, and the straight line they form is key to mastering algebra and beyond. It's like having the secret decoder ring to unlock a whole new world of mathematical possibilities! So, let's use this knowledge to tackle our table and complete those ordered pairs.

The Significance of the Slope

Let's dive a bit deeper into the equation y = (1/5)x to truly understand the relationship between x and y. This equation is in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. In our case, m is 1/5 and b is 0 (since there's no constant term added). The slope, 1/5, tells us how much y changes for every unit change in x. Specifically, for every 5 units x increases, y increases by 1 unit. This is crucial for visualizing the line and understanding its steepness. Think of it like climbing a hill; the slope tells you how steep the hill is. A slope of 1/5 means for every 5 steps you take horizontally, you climb 1 step vertically. The y-intercept, 0, tells us where the line crosses the y-axis. In this case, it crosses at the origin (0, 0). Knowing the slope and y-intercept gives us a powerful tool for quickly understanding and graphing linear equations. We can easily plot the y-intercept and then use the slope to find other points on the line. This understanding is not just about plugging in numbers; it's about grasping the geometric interpretation of the equation. So, when you see a linear equation, try to visualize the line in your mind, thinking about its slope and where it crosses the y-axis. This will make solving problems and understanding linear relationships much easier and more intuitive. Now, let's apply this deeper understanding to filling in our table and see how the slope plays out in the ordered pairs we find.

Completing the Table

Now, let's get to the fun part: completing the table of ordered pairs for the equation y = (1/5)x. We have a table with some x values and some y values, and our mission is to fill in the blanks. Remember, each row in the table represents an ordered pair (x, y) that satisfies the equation. This means if we plug the x value into the equation, we should get the corresponding y value, and vice versa. We'll go through each row one by one, using the equation to find the missing value.

Row 1: x = 0

For the first row, we're given x = 0. To find the corresponding y value, we simply substitute x = 0 into our equation: y = (1/5)(0). Anything multiplied by 0 is 0, so y = 0. This gives us the ordered pair (0, 0), which is the origin. Remember, we talked about the y-intercept earlier, and it was 0. This confirms that our line passes through the origin, which makes sense since b is 0 in our equation. The point (0, 0) is a crucial point on the line, and it's often the easiest to find because plugging in 0 for x usually simplifies the equation nicely. So, we've successfully completed the first row! Let's move on to the next one.

Row 2: x = -5

Next up, we have x = -5. Again, we substitute this value into our equation: y = (1/5)(-5). Multiplying a fraction by a whole number can sometimes feel tricky, but remember we're just multiplying the numerator (the top number) by the whole number. So, we have y = (-5/5), which simplifies to y = -1. This gives us the ordered pair (-5, -1). Notice that when x is negative, y is also negative. This makes sense because a positive fraction (1/5) multiplied by a negative number will always result in a negative number. Also, think about the slope we discussed earlier. For every 5 units x changes, y changes by 1 unit. Here, x decreased by 5 (from 0 to -5), so y decreased by 1 (from 0 to -1). This reinforces our understanding of the slope and how it relates the changes in x and y. We're on a roll! One more row to go.

Row 3: y = 1

This time, we're given y = 1 and need to find the corresponding x value. This means we'll substitute y = 1 into our equation: 1 = (1/5)x. To solve for x, we need to get rid of the (1/5) that's multiplying it. The easiest way to do this is to multiply both sides of the equation by the reciprocal of (1/5), which is 5. So, we have 5 * 1 = 5 * (1/5)x, which simplifies to 5 = x. This gives us the ordered pair (5, 1). This ordered pair perfectly illustrates the slope of our line. Remember, for every 5 units x increases, y increases by 1 unit. Here, x increased by 5 (from 0 to 5), and y increased by 1 (from 0 to 1). This connection between the slope and the ordered pairs helps solidify our understanding of linear equations. And just like that, we've completed the table!

The Completed Table and its Significance

Here's the completed table of ordered pairs for the equation y = (1/5)x:

These ordered pairs represent points that lie on the line represented by our equation. We can plot these points on a graph and connect them to visualize the line. This is a fantastic way to see the linear relationship between x and y in action. Moreover, these ordered pairs can be used for various applications, such as predicting values, solving systems of equations, and understanding real-world scenarios modeled by linear equations. Think about situations like calculating distance traveled at a constant speed, converting between units of measurement, or determining the cost of items based on a fixed price per item. All these scenarios can be represented by linear equations, and understanding how to find and interpret ordered pairs is key to solving them. So, completing this table wasn't just an exercise in plugging in numbers; it was a step towards unlocking the power of linear equations and their applications in the world around us!

Graphing the Linear Equation

Taking our understanding a step further, let's visualize the linear equation y = (1/5)x by graphing it. Graphing is a fantastic way to see the relationship between x and y and to confirm that our ordered pairs are correct. To graph the equation, we can use the ordered pairs we found in the table: (0, 0), (-5, -1), and (5, 1). First, we draw our coordinate axes, the x-axis (horizontal) and the y-axis (vertical). The point where the axes intersect is the origin (0, 0), which we already know is on our line. Next, we plot each of our ordered pairs. For (-5, -1), we move 5 units to the left along the x-axis and 1 unit down along the y-axis. For (5, 1), we move 5 units to the right along the x-axis and 1 unit up along the y-axis. Now, we have three points plotted on our graph. If we've done everything correctly, these points should form a straight line. Grab a ruler or straight edge and draw a line that passes through all three points. Extend the line beyond the points to show that it continues infinitely in both directions. Voila! We've graphed the linear equation y = (1/5)x. The line goes through the origin, which confirms our y-intercept is 0, and it has a gentle upward slope, reflecting the slope of 1/5. This graph is a visual representation of all the possible solutions to our equation. Any point on this line represents an ordered pair (x, y) that satisfies the equation. So, graphing isn't just a visual aid; it's a powerful tool for understanding and working with linear equations. And remember, the more we practice graphing, the more intuitive it becomes, helping us to quickly visualize and solve linear equations in various contexts.

Conclusion

Alright guys, we've covered a lot today! We started with the equation y = (1/5)x and learned how to complete a table of ordered pairs. We dove deep into understanding linear equations, ordered pairs, the significance of the slope and y-intercept, and even how to graph the equation. Remember, each ordered pair represents a point on the line, and the slope tells us how the line rises or falls. This is a fundamental concept in algebra and sets the stage for tackling more advanced topics. So, keep practicing, keep visualizing, and you'll become a master of linear equations in no time! The ability to work with linear equations is crucial in many areas of math and science, from solving problems in physics to understanding economic models. By mastering this foundational concept, you're setting yourself up for success in a wide range of fields. So, don't just memorize the steps; strive to understand the underlying principles. The more you grasp the 'why' behind the 'how,' the more confident and capable you'll become in your mathematical journey. Now, go out there and conquer those equations! You got this!