For the data in the table, does 𝑦 vary directly with 𝑥? If it does, write an equation for the direct variation.

Answer 1

To determine whether 𝑦 varies directly with 𝑥, we need to check if the ratio 𝑦/𝑥 remains constant for all the given values.

Let's calculate the ratio for each pair of 𝑥 and 𝑦:

For the first pair, 𝑦/𝑥 = 4/0.5 = 8
For the second pair, 𝑦/𝑥 = 6/0.75 = 8
For the third pair, 𝑦/𝑥 = 8/1 = 8
For the fourth pair, 𝑦/𝑥 = 10/1.25 = 8

Since the ratio 𝑦/𝑥 is constant (equal to 8) for all the values, we can conclude that 𝑦 does vary directly with 𝑥.

The equation for direct variation can be written as:

𝑦 = k𝑥

To find the value of k, we can use any pair of 𝑥 and 𝑦 from the table. Let's use the first pair (0.5, 4):

4 = k * 0.5

To isolate k, we divide both sides by 0.5:

k = 4 / 0.5
k = 8

Therefore, the equation for direct variation is:

𝑦 = 8𝑥