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Line 𝓁 has the equation y =

1
3
x. Find the equation of the image of 𝓁 after a dilation with a scale factor of
1
2
, centered at the origin.

If this mumble-jumble is supposed to say:

Line 𝓁 has the equation y = (1/3)x
Find the equation of the image of 𝓁 after a dilation with a scale factor of 1/2 , centered at the origin.

The given line already has a dilation factor of 1/3 of the line y = x
You are doing another of 1/2, so what do you think?

To find the equation of the image of line 𝓁 after a dilation with a scale factor of 1/2 centered at the origin, we can follow these steps:

1. Start with the equation of line 𝓁, which is y = (1/3)x.

2. Apply the dilation to the x-values and y-values of each point on the line:

- Multiply the x-values by the scale factor of 1/2.
- Multiply the y-values by the scale factor of 1/2.

3. Write the new equation using the transformed coordinates.

Let's apply these steps to find the equation of the image of line 𝓁 after the dilation:

1. Start with the equation of line 𝓁: y = (1/3)x.

2. Apply the dilation to the x-values and y-values:
- Multiply the x-values by 1/2: (1/2)x.
- Multiply the y-values by 1/2: (1/2)y.

3. Write the new equation using the transformed coordinates:
- The equation of the image of line 𝓁 after the dilation is y = (1/2)(1/3)x.

Simplifying this equation, we get:
- y = (1/6)x.

Therefore, the equation of the image of line 𝓁 after the dilation with a scale factor of 1/2 centered at the origin is y = (1/6)x.

To find the equation of the image of line 𝓁 after a dilation with a scale factor of 1/2 centered at the origin, we need to apply the dilation transformation to the equation of line 𝓁.

Let's start by considering a point (x, y) on line 𝓁. For any point on the line, the y-coordinate is equal to (1/3)x.

After the dilation with a scale factor of 1/2, the x-coordinate is halved, and the y-coordinate is also halved. So the new coordinates of the image point are (x/2, y/2).

Therefore, the equation of the image line is y' = (1/2)(1/3)x' = (1/6)x'.

Hence, the equation of the image of line 𝓁 after the dilation with a scale factor of 1/2 centered at the origin is y' = (1/6)x'.