To determine which of the given equations represents a linear relationship that is also a proportional relationship, we need to consider the definitions of these types of relationships.
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Linear relationship: An equation is linear if it can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. If \(b = 0\), the line passes through the origin (0,0).
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Proportional relationship: A proportional relationship is a specific type of linear relationship where the y-intercept \(b = 0\). Such an equation can be expressed in the form \(y = kx\), where \(k\) is the constant of proportionality.
Now, let's analyze the given equations:
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\(y = x^2\):
- This is a quadratic equation, not linear. It does not represent a linear relationship.
- Since it is not linear, it cannot be proportional either.
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\(y = 4x + 1\):
- This is a linear equation, where the slope \(m = 4\) and the y-intercept \(b = 1\).
- Since \(b \neq 0\), it does not pass through the origin and is not a proportional relationship.
In conclusion, neither of the given equations represents a linear relationship that is also a proportional relationship.
- Answer: Neither equation represents a linear relationship that is also a proportional relationship.