For each equation, determine whether
x
and
y
are directly proportional (that is, if the equation shows direct variation).
If so, then find the constant of proportionality (the constant of variation).
(a)
=y−x
Proportional
Constant of proportionality:
=k
Not proportional
(b)
=y75x
Proportional
Constant of proportionality:
=k
Not proportional
Note: An equation shows direct variation if it can be written in the form y = kx, where k is a constant of proportionality.
(a) y = k(x-1) - This equation is not in the form y = kx, so it does not show direct variation.
(b) y = (75)x - This equation can be written in the form y = kx by setting k = 75. Therefore, x and y are directly proportional with a constant of proportionality k = 75.
To determine if x and y are directly proportional in each equation, we need to see if they follow the form y = kx, where k is the constant of proportionality.
(a) y = x
In this equation, y and x are directly proportional as they have a linear relationship. The constant of proportionality is 1, so k = 1.
(b) y = 75x
In this equation, y and x are directly proportional as they have a linear relationship. The constant of proportionality is 75, so k = 75.
Therefore, for equation (a), x and y are directly proportional with a constant of proportionality of k = 1, and for equation (b), x and y are directly proportional with a constant of proportionality of k = 75.