To find the constant of proportionality between two variables \( x \) and \( y \), we typically focus on the ratios of these variables. The constant of proportionality \( k \) is often expressed in the context of a direct proportion, where \( y \) is proportional to \( x \) (i.e., \( y = kx \)).
Here's a breakdown of the methods you mentioned:
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Dividing \( y \) by \( x \): \[ k = \frac{y}{x} \] This gives you the constant of proportionality directly if \( y \) is proportional to \( x \).
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Dividing \( x \) by \( y \): \[ k = \frac{x}{y} \] This gives you the reciprocal of the constant of proportionality if you have \( x \) as proportional to \( y \) (i.e., \( x = ky \)).
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Multiplying \( x \) and \( y \): \[ k = xy \] This does not yield a constant of proportionality in the same way as division does. Instead, it gives a product of the two variables, which is not typically used for finding the constant of proportionality.
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Subtracting \( x \) and \( y \): \[ k = x - y \] This operation leads to a difference between the two variables and does not represent the constant of proportionality.
To summarize, the most relevant method to find the constant of proportionality is by dividing \( y \) by \( x \) if you have \( y = kx \) or dividing \( x \) by \( y \) if you have \( x = ky \). The other two operations do not yield the constant of proportionality in a conventional sense.
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