which sets of ordered pairs represents a linear relationship? if the relationship is linear, calculate the rate of change of the dependent variable.
1. (-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9)
2. (5,13),(6,15),(7,17),(8,19),(9,21),(10,23)
3. (1,-14),(3,-5),(6,4),(9,13),(12,22),(15,31)
If i'm correct 2 & 3 is linear. But i don't know how to calculate the rate of change of the dependent variable??
You are correct that both options 2 and 3 represent linear relationships. To calculate the rate of change of the dependent variable (also called the slope), you can use the formula:
Rate of change = (Change in the dependent variable) / (Change in the independent variable)
Let's calculate the rate of change for option 2:
For option 2:
x: 5, 6, 7, 8, 9, 10
y: 13, 15, 17, 19, 21, 23
Change in x: 6 - 5 = 1
Change in y: 15 - 13 = 2
Rate of change = (Change in y) / (Change in x) = 2/1 = 2
So, the rate of change for option 2 is 2.
Now let's calculate the rate of change for option 3:
For option 3:
x: 1, 3, 6, 9, 12, 15
y: -14, -5, 4, 13, 22, 31
Change in x: 3 - 1 = 2
Change in y: -5 - (-14) = 9
Rate of change = (Change in y) / (Change in x) = 9/2 = 4.5
So, the rate of change for option 3 is 4.5.
To determine if a set of ordered pairs represents a linear relationship, you can check if there is a constant rate of change between the dependent and independent variables.
Using this criteria, let's analyze each set of ordered pairs:
1. (-2,4),(-1,1),(0,0),(1,1),(2,4),(3,9)
In this set, the dependent variable (the second entry) does not have a constant rate of change. For example, between the first and second points, the dependent variable changes by -3. However, between the second and third points, the change is -1. Therefore, this set does not represent a linear relationship.
2. (5,13),(6,15),(7,17),(8,19),(9,21),(10,23)
In this set, the dependent variable increases by 2 for each increase of 1 in the independent variable. The rate of change is constant at 2. Therefore, this set represents a linear relationship.
3. (1,-14),(3,-5),(6,4),(9,13),(12,22),(15,31)
In this set, the dependent variable increases by 9 for each increase of 3 in the independent variable. The rate of change is constant at 9/3 = 3. Therefore, this set also represents a linear relationship.
So, you are correct that sets 2 and 3 represent linear relationships.
To calculate the rate of change of the dependent variable, you can choose any two points in the set. Let's take set 2 as an example:
Using (5,13) and (6,15), the change in the dependent variable is 15 - 13 = 2. With an increase of 1 in the independent variable, the dependent variable changes by 2. Therefore, the rate of change is 2.
Similarly, for set 3, you can choose any two points and calculate the change in the dependent variable over the change in the independent variable to find the rate of change.
divide the change in the dependent by the change in the independent.
For #2, that is 2/1,2/1,2/1,... xo it's linear
For #2, that's 9/2, 9/3, 9/3, 9/3, 9/3
They are all the same except for the first, so unless there's a typo, it's not linear.