Graphical Solution of Systems

Pre-Algebra · Axiom Academy

LESSON Graphical Solution of Systems Graph two equations on the same axes and the solution is staring back at you: the point where the lines cross. 1. The Crossing Point Is the Solution Take the system below. Each equation, on its own, is just a line. Graph both on one set of axes and watch where they meet — at (1, 2) the blue line and the red line pass through the same point , so that point lies on both. The intersection is the only point on both lines 2. One Solution: Different Slopes When the two lines have different slopes , they tilt at different angles, so they can only cross once. That single crossing is the system's one and only solution. Here y = 2x - 1 climbs steeply while y = -x + 5 falls — they meet at the grid point (2, 3) . 3. No Solution: Parallel Lines If the two lines have the same slope but different y -intercepts, they run side by side forever and never touch. With no crossing point, no (x, y) can satisfy both — the system has no solution . 4. Infinite Solutions: The Same Line Sometimes the two equations are secretly the same line in disguise. Divide 2y = 4x + 2 by 2 and it becomes y = 2x + 1 — identical to the first equation. The graphs land exactly on top of each other, so every point on the line solves both. Graphing turns a system into a picture: the solutions are wherever the two lines share points. Just read the slopes and intercepts and you can tell which of the three cases you're in. Same slope, different intercept Lines lie on top of each other

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