Final answer:
The correct response is that (x-1) and (y-1) are not in direct proportion because subtracting 1 from each variable alters the proportional relationship, affecting the original linearity and the constant of proportionality.
Step-by-step explanation:
The correct answer is option B) (x−1) and (y−1) are not in direct proportion. When two quantities are directly proportional, this means that if one quantity increases or decreases, the other quantity increases or decreases by the same proportion. This is represented by the equation y = kx, where k is the constant of proportionality. However, if we modify the variables to (x−1) and (y−1), the subtraction of 1 from each implies a shift and the initial linearity of the relationship is affected. This shift alters the equality of the ratio as the proportionality constant would no longer apply in the same way. For example, if x and y were originally equal (suggesting k = 1), subtracting 1 from both would result in two new quantities that are not equal, thus breaking the direct proportionality.
In direct proportion, when the value of one variable increases, the value of the other variable also increases at the same rate. Similarly, when the value of one variable decreases, the value of the other variable also decreases at the same rate. However, subtracting 1 from both x and y does not affect the direct proportionality between them.
For example, if x and y are directly proportional, and we have x = 3 and y = 6, then (x−1) = 2 and (y−1) = 5. Even though x and y are in direct proportion, (x−1) and (y−1) are not in direct proportion because the decrease in value for both x and y does not affect their relationship.