To derive the equation of a line through the origin in the form \(y = mx\), we need to identify the slope \(m\) of the line. The slope \(m\) can be found by determining how much \(y\) changes for a given change in \(x\) (rise over run).
Based on the responses provided, we can evaluate the options. For a line through the origin, it should have the form \(y = mx\), where \(m\) is the slope.
From the responses:
- \(y = x\) implies a slope of 1.
- \(y = 8x\) implies a slope of 8.
- \(y = 33x\) implies a slope of 33.
- \(y = 3x\) implies a slope of 3.
Based on these, if we choose \(y = 3x\) and simplify it, we get:
\[ y = 3x \] This is already in its simplest form.
Among the given options, you can choose any equation depending on the slope, but the most simplified version of an equation through the origin representing a direct proportionality would be \(y = mx\).
For the specific cases provided:
- If \( m = 1 \): \( y = x \) (which can also be written as \(y = 1x\))
- If \( m = 8 \): \( y = 8x \)
- If \( m = 33 \): \( y = 33x \)
- If \( m = 3 \): \( y = 3x \)
All these equations represent a line through the origin with different slopes. Without additional context or specific graph details, any of these equations could be considered correct based on different slopes. However, if simply asked for a general form, you would represent a line through the origin as:
Final Answer: \(y = mx\) (specific \(m\) depending on context.)