the line whose equation is y=1/2x-3 is dilated by a factor of 3 with a center at the point (10,2). Explain why its equation would not change.

we  have that

y=1/2x-3
1) the slope of this line equation is m=(1/2)
2) the coordinates of the y intercept are
for x=0
y=-3
point (0,-3)
3)the coordinates of the x intercept are
for y=0
0=(1/2)x-3------> (1/2)x=3----> x=6
point (6,0)

if the line is dilated by a factor of 3
then
y=1/2x-3--------> 3y=3*[1/2x-3]-----> 3y=(3/2)x-9
3y=(3x-18)/2-------> y=(3x-18)/6--------> y=(3/6)x-18/6

 y=(3/6)x-18/6
so
1) the slope of this line equation is m=(3/6)-----> 3/6=1/2
2) the coordinates of the y intercept are
for x=0
y=-18/6-------> y=-3
point (0,-3)
3)the coordinates of the x intercept are
for y=0
0=(3/6)x-18/6------> (3/6)x=18/6----> x=18/3------> x=6
point (6,0)

therefore
the equation of the line
y=1/2x-3 is the same that the equation of the line  y=(3/6)x-18/6

the equation does not change because the slope and the x and y intercepts do not change either