A line does not care how you write it. Slope-intercept, point-slope, and standard form are three sentences describing the same set of points — each one putting a different fact in plain sight. Move the sliders and watch all three rewrite themselves at once, then step through the algebra that turns any one of them into another.
One line
The same line, three ways
Rise — vertical change3
−6 falling0 flat6 climbing
Run — horizontal change2
0 vertical6
A point it passes through — x14
−606
… and y1−3
−606
Or jump to a line worth seeing
Which conversion?
Why rewriting is allowed at all. Adding the same number to both sides, or multiplying both sides by the same nonzero number, never changes which pairs (x, y) make the sentence true. The solution set is untouched — and the solution set is the graph. So the picture cannot move, no matter how the writing is rearranged. Notice that the line above never budges while the three forms rewrite themselves.
The three traps.One: in point-slope, the sign of the point gets subtracted, so the point (4, −3) gives y − (−3), which tidies to y+ 3 — the sign you see is the opposite of the coordinate. Two: an equation is only in slope-intercept form once it is solved for y alone; 2y = 6x + 4 has not got there yet, and reading its slope as 6 is wrong. Three: from standard form the slope is −A/B, not A/B.