The equation of the line a is y = -2x + 5. Find the equation of the image of line a after a dilation centered at the origin with scale factor 2.

-4x + 10?

An equation of a line must include an equal sign, and your answer does not have one.

You probably meant
y=-4x+10

Recall some properties of a dilation includes
1. conservation of parallelism,
------------------------------
i.e. lines before and after dilation should remain parallel.
We see that
y=-4x+10 has double the slope of
y=-2x+5, which means that
the former is not the dilalted line.
2. OP'/OP = factor of dilation
Lengths of line joining the centre of dilation (O) with any point P before and after dilation (P') should be in a ratio equal to the factor of dilation.

For example, let P(0,5) be on the original line, then Length O-P = 5.
the point P' should have a length of 10 while staying on the y-axis, i.e. P'(0,10), or length O-P'=10.
Thus a line with slope -2 passing through (0,10) is
y=-2x+10 would be the dilated line.

To find the equation of the image of line 'a' after a dilation, we need to apply the dilation transformation to the equation of line 'a'.

The equation of line 'a' is given as y = -2x + 5.

In a dilation centered at the origin with a scale factor of 2, the x-coordinate is multiplied by 2 and the y-coordinate is also multiplied by 2.

Thus, the new equation of the line after the dilation can be found by substituting 2x for x and 2y for y in the equation of line 'a'.

Replacing x with 2x and y with 2y, we get:
2y = -2(2x) + 5

Simplifying further:
2y = -4x + 5

Hence, the equation of the image of line 'a' after the dilation is y = -4x + 5.

To find the equation of the image of line a after a dilation centered at the origin with a scale factor of 2, we need to multiply the coefficients of x and y by the scale factor.

Given that the equation of line a is y = -2x + 5, we can apply the dilation by multiplying the coefficients of x and y by 2:

New coefficient of x = -2 * 2 = -4
New coefficient of y = 1 * 2 = 2

Therefore, the equation of the image of line a after the dilation is y = -4x + 10.

So, your answer of -4x + 10 is correct!