Assuming x - a = y - a and x + 7 = y + 7
Original Point P (4, -2)
Translated to Point P' (x + 3, y - a) = (4 + 3, -2 - a) = (7, -2 - a)
Translated to next point P'' = (x - b, y + 7) = (7 - b, -2 - a + 7) = (7 - b, 5 - a) = (-5, 8)
From the above changes, we can see that 7 - b = -5 and 5 - a = 8. Therefore:
[tex]\begin{gathered} 7-b=-5 \\ 7+5=b \\ 12=b \end{gathered}[/tex][tex]\begin{gathered} 5-a=8 \\ 5-8=a \\ -3=a \end{gathered}[/tex]
The value of a = -3 and b = 12.
The point P' (7, -2 - a) = (7, -2 - (-3)) = (7, 1). Point P' is at (7, 1).
To check if this is right, let's look at the original point again and its transformations.
P (4, -2) translated to (x + 3, y - a) = (4 + 3, -2 - (-3)) = (7, 1).
P' (7, 1) is then translated to ( x - b, y + 7) = (7 - 12, 1 + 7) = (-5, 8).
As mentioned in the question, P'' is indeed found at (-5, 8).
