Question

For each function, determine whether Y varies directly with X. If so, find the constant of variation and write the function rule. Show your step-by-step work.

X Y
2 14
3 21
5 25

Answer 1

To determine if Y varies directly with X, we need to check if there's a constant ratio between Y and X for each pair of values. If there is, we can find the constant of variation and write the function rule.

Let's calculate the ratios for each pair of values:

For the first pair (X = 2, Y = 14):
Y/X = 14/2 = 7
For the second pair (X = 3, Y = 21):
Y/X = 21/3 = 7
For the third pair (X = 5, Y = 25):
Y/X = 25/5 = 5
The ratios for the first two pairs are the same (7), which indicates that Y varies directly with X. However, the ratio for the third pair is different (5).

So, Y varies directly with X for the first two pairs. Let's find the constant of variation (k) and write the function rule using the first pair:

From the first pair (X = 2, Y = 14), we have:

k = Y/X = 14/2 = 7

Now, we can write the function rule:

Y = kX

Substitute the value of k:

Y = 7X

Therefore, the function rule for this relationship is Y = 7X when Y varies directly with X.