Which table of values is a linear relationship? Write an equation for the relationship.
x|5 10 15 20 25 30
y|20 30 40 50 50 50
I want to say that x is but I have no theory as to why. I have looked at sample problems in the text and still do not understand how to do this. Any help would be appreciated!!
Well, let me break it down for you, with a touch of clowning around, of course!
If a relationship is linear, it means that the change in y is consistent for each change in x. In other words, as x increases by a certain amount, y should increase by the same amount every time. Let's take a look at your table.
x|5 10 15 20 25 30
y|20 30 40 50 50 50
If we calculate the change in y for each step, we can figure out if it's consistent. So, let's have some fun with math:
Change in y from 5 to 10: 30 - 20 = 10
Change in y from 10 to 15: 40 - 30 = 10
Change in y from 15 to 20: 50 - 40 = 10
Change in y from 20 to 25: 50 - 50 = 0
Change in y from 25 to 30: 50 - 50 = 0
Oh dear, oh dear! It seems like the change in y is not consistent. The first three changes are indeed consistent (10, 10, 10), but the last two changes are zero.
Now let's have a blast at writing the equation for the relationship. We can use the formula for finding the slope of a line, which is (change in y) / (change in x).
With our table, the change in y from 5 to 15 is 40 - 20 = 20, and the change in x is 15 - 5 = 10. So the slope is 20 / 10 = 2.
To find the y-intercept, we can plug in any x and y values from the table into the equation y = mx + b and solve for b. Let's use x = 5 and y = 20:
20 = 2(5) + b
20 = 10 + b
b = 10
So, the equation for the linear relationship is y = 2x + 10. Ta-da!
Remember, clowning around with math can make it more enjoyable and less scary. I hope this helps you understand linear relationships a little better. If you need more laughs or explanations, I'm here for you!
To determine if a table of values represents a linear relationship, you can observe if there is a consistent rate of change between the x and y values. In a linear relationship, the difference in the y-values (Δy) is constant for every corresponding change in the x-values (Δx).
In the given table, let's calculate the differences between consecutive x and y values:
x|5 10 15 20 25 30
y|20 30 40 50 50 50
Δx: 10-5=5, 15-10=5, 20-15=5, 25-20=5, 30-25=5
Δy: 30-20=10, 40-30=10, 50-40=10, 50-50=0, 50-50=0
As we can see, the Δx values are consistent at 5, which indicates a constant interval between x-values. However, the Δy values are not consistent. You can see that Δy is 10 for the first three values but then becomes 0 for the last three values.
Since there is not a consistent rate of change between the y-values, the given table does not represent a linear relationship.
As for writing an equation for a linear relationship, it is typically in the form of y = mx + b, where m is the slope (rate of change) and b is the y-intercept. However, since the given table does not represent a linear relationship, we cannot determine an equation for it.