Explanation:
these are very simple situations that experienced people do normally in their head.
I will do my best to describe the line of thinking :
starting with "a".
since left and right side of the equal sign are supposed to be equal, we can say that every part of the left and the right side must satisfy the equal relationship.
and because every other term is indeed equal, there is no other "choice", (6-2x) must be equal to a(2x-6) to maintain the equality.
6 - 2x = a(2x - 6) = a×-1×(6 - 2x)
1 = a×-1
a = -1
now for "b".
9/(a(2x-6) = b/(2x-6)
9/(-1×(2x-6)) = b/(2x-6)
9/-1 = b
b = -9
now for "c".
(3x - c)/(2x - 6) = 3x/(2x - 6) + b/(2x - 6) =
= 3x/(2x - 6) + -9/(2x - 6) =
= 3x/(2x - 6) - 9/(2x - 6) =
= (3x - 9)/(2x - 6)
therefore,
3x - c = 3x - 9
-c = -9
c = 9
now for "d", "e" and "f".
d×(x - e) = 3x - c = 3x - 9
now, what do we need to do to get an x-term out of a 3x- term ? we "pull out" the 3 as factor of the term.
3x - 9 = 3×(x - 3)
so,
d×(x - e) = 3×(x - 3)
and we see
d = 3
e = 3
f×(x - 3) = 2x - 6
the same line of thinking : how to get an x- term out of a 2x-term ? we "pull out" 2 as factor of the term.
2x - 6 = 2×(x - 3)
so,
f×(x - 3) = 2×(x - 3)
f = 2
now for "g".
d×(x - e)/(f×(x - 3)) = g
3×(x - 3)/(2×(x - 3)) = 3/2 = g = 1.5