For which table(s) of values in Exercises 39–42 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
I do not think that this is a linear equation but I cannot determine the pattern so that I know why it isn't linear.
NOt linear. notice how y becomes constant at high x, that is not linear as the first.
To determine if a relationship is linear, we need to check if there is a constant rate of change between the variables. In this case, we can calculate the differences between consecutive y-values and x-values:
x: 5 10 15 20 25 30
y: 20 30 40 50 50 50
Δy: 10 10 10 0 0
Δx: 5 5 5 5 5
Calculating the differences, we see that the changes in y (Δy) are not constant. For example, there is a change in y of 10 between the first and second pair of values, but only a change of 0 between the fourth and fifth pair of values. Since the rate of change is not constant, the relationship is not linear.
To find the equation for a linear relationship, we need a constant rate of change. Since this relationship is not linear, we cannot write a linear equation for it.
To determine if a relationship is linear, you need to examine the pattern and see if there is a constant rate of change between the x and y values.
In this case, let's calculate the differences between consecutive y-values:
Difference between 20 and 30: 30 - 20 = 10
Difference between 30 and 40: 40 - 30 = 10
Difference between 40 and 50: 50 - 40 = 10
Difference between 50 and 50: 50 - 50 = 0
As you can see, the difference between consecutive y-values is not constant. In a linear relationship, the difference between the y-values should remain the same. Since the differences are not constant in this case, we can conclude that the relationship is not linear.
To write the equation for a linear relationship, we require a constant rate of change, which we do not have here. Therefore, it is not possible to provide an equation for a linear relationship for this table of values.