Shifting of Origin

When the origin is shifted to (h, k), new coordinates (X, Y) relate to old by x = X + h, y = Y + k.

Let the origin shift to $O'(h,k)$. A point $P$ with old coordinates $(x,y)$ has new coordinates $(X,Y)$ relative to $O'$.

Since $P = O' + (X,Y)$:

$$x = X + h, \qquad y = Y + k$$

or:

$$X = x - h, \qquad Y = y - k$$

Effect on a line

$ax + by + c = 0$ becomes $a(X+h) + b(Y+k) + c = 0$, i.e.:

$$aX + bY + (ah+bk+c) = 0$$

Slope is unchanged. If $(h,k)$ is chosen as the intersection of two lines, both simplify simultaneously — the standard use in pair-of-lines problems to eliminate linear terms.

What is preserved

Slopes, angles between lines, distances between points, the degree of any equation, the form of any equation.

What changes

Intercepts, coordinates of all points, the constant term in line equations.