To derive the equation \( y = mx \) for a line through the origin, we can interpret \( m \) as the slope of the line.
From the options you provided, we can break them down as follows:
- \( y = 3x \): This means the slope \( m = 3 \).
- \( y = 8x \): This means the slope \( m = 8 \).
- \( y = \frac{3}{3} x \): This simplifies to \( y = 1x \), which means the slope \( m = 1 \).
- \( y = x \): This is the same as \( y = 1x \), meaning the slope \( m = 1 \).
Thus, the simplest forms of the equations \( y = mx \) can be summarized as:
- For \( m = 3 \), the equation is \( y = 3x \).
- For \( m = 8 \), the equation is \( y = 8x \).
- For \( m = 1 \), the equation is \( y = x \) or \( y = \frac{3}{3}x \) (which is likewise \( y = x \)).
The simplest and most fundamental form for a line that passes through the origin is \( y = mx \), where \( m \) is a constant. The numerical coefficients given \( 3, 8, \) or \( 1 \) represent different slopes of the line.
If the question is about identifying lines through the origin from your selection based on slope, all provided equations represent lines through the origin, but \( y = x \) (or \( y = \frac{3}{3}x \)) is the simplest one with a slope of \( 1 \).