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
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