Determine whether y varies directly with x. Find the constant of variation. Write the function rule (the equation)

X: -2,1,4
Y: 6,-3,-12

y varies directly with x if

y = kx
for some constant k.

That is, y/x is a constant value
so try the values given

Thank you!

To determine whether y varies directly with x, we need to check if there is a constant ratio between y and x.

First, let's calculate the ratio between y and x for each given pair of values:

For the first pair, when x = -2 and y = 6, the ratio is y / x = 6 / (-2) = -3.
For the second pair, when x = 1 and y = -3, the ratio is y / x = (-3) / 1 = -3.
For the third pair, when x = 4 and y = -12, the ratio is y / x = (-12) / 4 = -3.

Since the ratio between y and x is the same (-3) for all three pairs, we can conclude that y varies directly with x.

The constant of variation (k) is the constant ratio between y and x. In this case, k is -3.

The function rule (equation) for this direct variation is:

y = kx

Substituting the value of k, we have:

y = -3x

To determine whether y varies directly with x, we need to check if there is a constant ratio between the values of x and y.

Step 1: Calculate the ratio between any pair of x and y values.
If we choose the first pair (-2, 6), the ratio is:

(6) / (-2) = -3

If we choose the second pair (1, -3), the ratio is:

(-3) / (1) = -3

If we choose the third pair (4, -12), the ratio is:

(-12) / (4) = -3

Step 2: Compare the ratios.
Since the ratio is the same for all three pairs, we can conclude that y varies directly with x.

Step 3: Find the constant of variation.
The constant of variation represents the ratio between y and x. In this case, the ratio is -3.

Step 4: Write the function rule (equation).
The function rule (equation) for the relationship between x and y, where y varies directly with x with a constant of variation of -3, is:

y = -3x