Final answer:
The equation of the image line after dilation by a scale factor of 1/3 from the origin is y = 2/3x.
Step-by-step explanation:
When dilating a line from the origin by a scale factor of k, the equation of the image line can be found by multiplying the original equation by the scale factor k. In this case, the line y=2x is being dilated by a scale factor of 1/3, so the equation of the image line is obtained by multiplying the original equation by 1/3: The original equation of the line is y = 2x. When a line is dilated by a scale factor from the origin, every x and y coordinate of points on the line is multiplied by that factor. In this case, the scale factor is 1/3. Therefore, the new equation of the line after dilation will be the original equation with both x and y multiplied by 1/3, yielding the image equation y = (1/3)(2x). Simplifying this we get y = 2/3x, which is the equation of the line after the dilation.