To answer this question we first need to understand what it means for y to vary directly with x. Direct variation describes a simple relationship between two variables. If y varies directly with x, it means that when x increases, y increases by the same factor, and similarly, when x decreases, y decreases by the same factor.
We also need to understand the constant of variation, denoted as k. This is the factor by which x and y increase or decrease together. If y = kx, we can say that y varies directly with x, and k is the constant of variation.
Let's take the given values y = 12 and x = 3.
For option a, if the constant of variation k = 4, then substituting values into the equation y = kx we get y = 4 * 3. After performing the multiplication we get y = 12.
Since our calculated y value is equal to our given y value (both 12), we can conclude that y does vary directly with x with the constant of variation k = 4. So, the correct answer is option a: yes; k = 4; y = 4x.