Every point on a line is a solution of that line’s equation. So a point that solves two equations at once must sit on two lines at once — and there it is, the whole idea of solving a system by graphing. What makes it worth a whole section is that this leaves room for exactly three endings, never a fourth. Drag either line below and see which ending you land in.
Two lines, one plane
Three endings, and nothing else
Compare the two slopes before you draw a single point. If the slopes differ, the lines lean differently and must cross — exactly once, no matter how far out that crossing sits. If the slopes match but the intercepts do not, the lines climb in perfect lockstep at different heights and never touch. If slope and intercept both match, you were handed one line wearing two disguises.
The vocabulary that goes with the three pictures. A system with at least one solution is consistent; one with no solution is inconsistent. Among the consistent ones, exactly one solution makes the equations independent, and infinitely many makes them dependent — the second equation was carrying no new information.
Three ways this section is misread. 1. “No solution” is not “the solution is 0.” Zero is a perfectly good number, and (0, 0) would be a real answer. No solution means the solution set is empty — there is no ordered pair anywhere in the plane that satisfies both equations. 2. “Infinitely many” is not “anything works.” The solutions are exactly the points on that one shared line. For y = −x + 4, the pair (0, 0) is not a solution: −0 + 4 = 4, not 0. 3. A graph is a picture, and pictures estimate. If the crossing does not land on a grid corner, reading it off gives a near-miss — and two nearly-parallel lines can shove the true crossing right off the page. This is exactly why substitution and elimination exist. Always substitute your answer back into both original equations. One more form to expect. A vertical line such as x = 4 has no y = mx + b form at all, so the slope comparison above has nothing to compare. It still graphs perfectly well, and a system containing one is read off the picture the same way.
drag a line to begin
Equation 1 — the solid line
slope m11
intercept b11
Equation 2 — the dashed line
slope m2−1
intercept b23
Drag any of the four dots straight on the grid — the round dot slides a line up and down, the square dot tilts it. Or nudge a slider with the arrow keys.