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 y

5 20
10 30
15 40
20 50
25 50
30 50

Look at y. It stays constant at the end. If it were always constant, it would be linear...a horizontal line with zero slope. But near the origin, it is is not constant.

The relation is not linear.

Thank you! :)

So, no part of the table is linear? I mean, the x=5, y=20; x=10, y=30; & x=15, y=40, none of those are linear either? Or is it because of the constant at the end that none of it is linear?

To determine if the relationship between the given values is linear, we need to check if the ratio of the change in y to the change in x is constant.

For example, let's calculate the change in y and x for each set of values:

- For the first set of values:
- Change in y: 30 - 20 = 10
- Change in x: 10 - 5 = 5
- Ratio: 10/5 = 2

- For the second set of values:
- Change in y: 40 - 30 = 10
- Change in x: 15 - 10 = 5
- Ratio: 10/5 = 2

- For the third set of values:
- Change in y: 50 - 40 = 10
- Change in x: 20 - 15 = 5
- Ratio: 10/5 = 2

- For the fourth set of values:
- Change in y: 50 - 50 = 0
- Change in x: 25 - 20 = 5
- Ratio: 0/5 = 0

- For the fifth set of values:
- Change in y: 50 - 50 = 0
- Change in x: 30 - 25 = 5
- Ratio: 0/5 = 0

- For the sixth set of values:
- Change in y: 50 - 50 = 0
- Change in x: 30 - 25 = 5
- Ratio: 0/5 = 0

Since the ratio of the change in y to the change in x is constant (2) for the first three sets of values, we can conclude that the relationship is linear.

To write the equation for each linear relationship, we can use the slope-intercept form of a linear equation, y = mx + b, where m represents the slope and b represents the y-intercept.

Using the first set of values (5, 20) and (10, 30) to find the slope:
m = (y2 - y1) / (x2 - x1)
= (30 - 20) / (10 - 5)
= 10 / 5
= 2

Now that we have the slope, we can substitute it into the equation along with one set of coordinates (5, 20):
20 = 2 * 5 + b

Simplifying the equation:
20 = 10 + b

Solving for b:
b = 20 - 10
b = 10

Therefore, the equation for the linear relationship is y = 2x + 10.