A linear equation shows a proportional relationship if it can be written in the form \( y = kx \), where \( k \) is a constant. This means that the line passes through the origin (0,0).
Let's analyze the given equations:
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\( y = -\frac{1}{6}x \): This equation is in the form \( y = kx \) with \( k = -\frac{1}{6} \), which means it is proportional.
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\( y = \frac{1}{6}x - 8 \): This equation has a y-intercept of -8, which means it does not pass through the origin. Therefore, it is not proportional.
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\( y = -6x + 1 \): This equation has a y-intercept of 1, which means it does not pass through the origin. Therefore, it is not proportional.
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\( y = 6 \): This is a horizontal line that does not depend on \( x \) and does not show a direct proportional relationship. It represents a constant value, which means it is not of the form \( y = kx \).
The only linear equation that shows a proportional relationship is:
\( y = -\frac{1}{6}x \).