I don't think I understand what is meant by "to shift the origin of coordinates to the point $(h,k)$ in coordinate geometry. I've read Loney's book on coordinate geometry in which he says that to shift the origin of coordinates that a curve is referred by, you must replace $x$ with $x + h$ and $y$ by $y + k$. Also, in some online examples I seem to be utterly puzzled. Here's an example I found on unacademy.com:
Example 1: Find the new equation of the following curve after the coordinates are transformed: $x + 3y = 6$, when the origin is transferred to the unique point $(–4, 1)$.
they replace $x$ with $x-4$ and $y$ with $y + 1$ to get $x + 3y = 7$. However, using an online graphing calculator, I can see that it has actually been shifted to $(4,-1)$. Hence, I think I misunderstand what is meant by "shift of origin". the original graph $x + 3y = 6$ was two units up the y-axis from the original origin, hence the new graph $x + 3y = 7$ should also be two units up the y-axis from the new origin, however it is not. Could someone please help me, I am really stuck and cannot further explore the subject. Thank you in advance.