Graphing System Solutions

Algebra 1 · Axiom Academy

LESSON Graphing System Solutions Two lines, one grid: where they cross is the single pair (x, y) that solves both equations at once. 1. The Solution Is Where the Lines Cross Each line is the full set of points that make its own equation true. A solution to the system has to satisfy both equations — so it must sit on both lines at the same time. Graph both, and the only such point is the spot where they intersect . Line 1 — every point making the first equation true Line 2 — every point making the second equation true Take one equation at a time. In slope-intercept form y = mx + b , the number b tells you where the line crosses the y -axis, and the slope tells you how to step to the next point. For y = -x + 5 , start at b = 5 , then go down 1, right 1 to land the next point — and connect them. The y -intercept is b = 5 , so put a point at (0, 5) . Slope : from (0,5) go down 1, right 1 to (1, 4) . 3. Add the Second Line and Read the Crossing Now graph y = 2x - 1 on the same grid : intercept at (0, -1) , slope so step up 2, right 1. Where the two lines meet is the solution. Drop a line straight down to the x -axis to read x , and straight across to the y -axis to read y . Reading a graph can be off by a little, so confirm by substituting. The true solution makes both equations true. A point on only one line — like (1, 1) , which sits on Line 2 but not Line 1 — is not a solution of the system, and substitution catches it. Substitute (2,3) into Line 1: 3 = -(2) + 5 = 3 — true.

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