What is the value of the constant of variation for the direct variation that goes through (4, 12)?

In a direct variation, the relationship between two variables, often denoted as ( x ) and ( y ), can be expressed in the form ( y = kx ), where ( k ) is the constant of variation. To find the constant of variation, we can use the coordinates of the point given in the problem.

Given the point (4, 12), we can substitute these coordinates directly into the equation. Here, ( x = 4 ) and ( y = 12 ). Plugging these values into the equation gives us:

[

12 = k \cdot 4

]

To solve for ( k ), we can rearrange the equation:

[

k = \frac{12}{4}

]

Calculating this yields:

[

k = 3

]

Thus, the constant of variation is 3. This indicates that for every increase of 1 in ( x ), ( y ) increases by a factor of 3. This is why the answer is the value of ( k ) which represents the constant relationship in this direct variation scenario.