1. The problem asks to identify which equations show $y$ in a proportional relationship with $x$. 2. A proportional relationship means $y$ is directly proportional to $x$, which can be written as $y = kx$ where $k$ is a constant. 3. Let's analyze each equation: - $y = 5x$: This is $y = kx$ with $k=5$, so it is proportional. - $y = \frac{1}{5} x$: This is $y = kx$ with $k=\frac{1}{5}$, so it is proportional. - $y = 5 - x$: This is $y = -x + 5$, not of the form $kx$, so not proportional. - $y = \frac{1}{5} + x$: This is $y = x + \frac{1}{5}$, not of the form $kx$, so not proportional. - $y = \frac{1+x}{5}$: This is $y = \frac{1}{5} + \frac{x}{5}$, not of the form $kx$, so not proportional. - $y = \frac{5}{1+x}$: This is not linear and not of the form $kx$, so not proportional. 4. Therefore, the equations showing proportional relationships are: $$y = 5x \quad \text{and} \quad y = \frac{1}{5} x$$ Final answer: $y = 5x$ and $y = \frac{1}{5} x$ show proportional relationships with $x$.