Is this relationship linear? Tell how you decided. Write the equation for each linear relationship.

X 1, 2, 5, 8, 10, 11
Y 1.5, 2.5, 5.5, 8.5, 10.5, 11.5

To determine whether the given relationship is linear, we need to examine the pattern between the x-values and y-values.

We can start by plotting the points on a coordinate plane:

(X, Y) = (1, 1.5), (2, 2.5), (5, 5.5), (8, 8.5), (10, 10.5), (11, 11.5)

After plotting the points, we can observe that they lie more or less in a straight line. However, to confirm if the relationship is indeed linear, we can calculate the difference in y-values for each pair of consecutive x-values:

For (1, 1.5) and (2, 2.5): Δy = 2.5 - 1.5 = 1
For (2, 2.5) and (5, 5.5): Δy = 5.5 - 2.5 = 3
For (5, 5.5) and (8, 8.5): Δy = 8.5 - 5.5 = 3
For (8, 8.5) and (10, 10.5): Δy = 10.5 - 8.5 = 2
For (10, 10.5) and (11, 11.5): Δy = 11.5 - 10.5 = 1

Upon examining the differences in y-values, we can conclude that they are not constant. In a linear relationship, the difference in y-values should remain constant for every pair of consecutive x-values. Therefore, we can determine that the given relationship is not linear.

However, if we were to assume a linear relationship, we can calculate the equation using the formula for the equation of a straight line:

y = mx + b

To find the slope (m), we can select two points and calculate the difference in y-values (Δy) divided by the difference in x-values (Δx):

m = Δy / Δx

Choosing the points (1, 1.5) and (5, 5.5), we have:

m = (5.5 - 1.5) / (5 - 1) = 4 / 4 = 1

Now, we can select one of the points and substitute it into the equation along with the slope to solve for the y-intercept (b):

1.5 = (1)(1) + b
b = 1.5 - 1
b = 0.5

The equation for the assumed linear relationship is:

y = x + 0.5

However, it's essential to note that this equation is hypothetical, as we determined the relationship to be nonlinear based on the inconsistent differences in y-values.