Question

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)   
=−45xy
Proportional
Constant of proportionality:
=k
  
Not proportional
(b)   
=y+54x
Proportional
Constant of proportionality:
=k
  
Not proportional

Answer 1

To determine if the variables \( x \) and \( y \) are directly proportional based on the given equations, we need to see if they can be expressed in the form \( y = kx \) where \( k \) is a constant.

**(a)** Given the equation:
\[
y = -45xy
\]

We can rearrange this equation:
\[
y + 45xy = 0 \implies y(1 + 45x) = 0
\]
This implies either \( y = 0 \) or \( 1 + 45x = 0 \).

Since it's not of the form \( y = kx \) for a constant \( k \) (because the equation depends on \( x \) in a non-linear way), we can conclude that:

- **Not proportional**

**(b)** Given the equation:
\[
y = y + 54x
\]
Rearranging the terms gives:
\[
y - y = 54x \implies 0 = 54x
\]
This means \( x \) must be 0 if the equation holds true, which does not establish a proportional relationship.

Thus, the equation does not match the direct proportionality condition either.

- **Not proportional**

In conclusion:
(a) Not proportional
(b) Not proportional