For which values is the relationship linear? Tell how you decided. Write the equation for each linear relationship
x 5 10 15 20 25 30
y 20 30 40 50 50 50
It is linear from x = 5 to x = 20 because the ratio of the change in y to the change in x is a constant over that interval. There is a different linear relationship (y = constant)for 20<x<30.
The slope is discontinuous at x=20. Two linear lines join there (if you were to plot a graph).
I still can't quite grasp it but your explanation is very helpful. Thank you!
To determine if the relationship between x and y is linear, we need to check if there is a constant rate of change between the x-values and the corresponding y-values.
First, let's calculate the rate of change between each pair of consecutive x-values and y-values:
Between x=5 and x=10, the change in x is 10-5 = 5, and the change in y is 30-20 = 10. The rate of change is 10/5 = 2.
Between x=10 and x=15, the change in x is 15-10 = 5, and the change in y is 40-30 = 10. The rate of change is 10/5 = 2.
Between x=15 and x=20, the change in x is 20-15 = 5, and the change in y is 50-40 = 10. The rate of change is 10/5 = 2.
Between x=20 and x=25, the change in x is 25-20 = 5, and the change in y is 50-50 = 0. The rate of change is 0/5 = 0.
Between x=25 and x=30, the change in x is 30-25 = 5, and the change in y is 50-50 = 0. The rate of change is 0/5 = 0.
We can observe that the rate of change (or slope) of the relationship is 2 for the first three pairs of values (x=5,10,15), but it becomes 0 for the last two pairs (x=20,25 and x=25,30). Since a linear relationship has a constant rate of change, we can conclude that the relationship is linear only for x=5,10,15.
Now, let's find the equation for the linear relationship:
Since the slope is constant and equal to 2 for x=5,10,15, we can choose any of these points to write the equation. Let's use x=5.
Using the point-slope form of a linear equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, we have:
y - 20 = 2(x - 5)
y - 20 = 2x - 10
y = 2x + 10
So, the equation for the linear relationship is y = 2x + 10.