Y is directly proportional with x. Write an equation that shows the relationship if x = 2 and y = 6?

Direct proportionality is defined as:

#y = alpha x#

where #alpha# is some constant that defines the proportionality.

Given #x = 2# and #y = 6#, we find:

#y = alpha x#

#6 = alpha (2)#

#3 = alpha#

So the relationship here is #y = 3x#

When two variables are directly proportional, it means that one is a constant multiple of the other. For example, in the equation #y = 16x#, #y# is directly proportional to #x#, because #y# is just some constant multiple of #x#. (In this case, the constant multiple is 16.)

The equation #y = x^2# does not represent a directly proportional relationship, because #y# is not some constant multiple of #x#.

To the problem at hand -- we are given that #y# and #x# are directly proportional. This means #y# is a constant multiple of #x#. This can be written as #y = kx#, where #k# is some constant multiple (a number).

We have the equation #y = kx# and we are also told that #x = 2# and #y = 6#. We can directly plug these in to determine the value of #k#. #y = kx -> 6 = 2k -> k = 3#. Thus, our relationship is given by the equation #y = 3x#. This is our final answer.