Solving systems by graphing

The point (2, −1) makes both 3x − y = 7 and x − 2y = 4 true, so it is where their two lines cross.

Two equations, one ordered pair

A system of linear equations is two or more equations considered together. A solution of a system of two equations is an ordered pair (x, y) that makes both equations true at the same time. To check a pair, substitute it into each equation. OpenStax uses the system 3x − y = 7 and x − 2y = 4 with the pair (2, −1): 3(2) − (−1) = 7 and 2 − 2(−1) = 4, so it works for both and is a solution. A pair that satisfies only one of the equations is not a solution of the system.

Graphing turns this into a picture. Each equation has a line made of all its solutions. A point on both lines is a solution of both equations, so the solution of the system is the point where the lines intersect. To solve by graphing, write each equation in slope-intercept form, graph both lines on the same grid, and read the intersection. For the OpenStax system, 3x − y = 7 becomes y = 3x − 7 and x − 2y = 4 becomes y = (1/2)x − 2. Drawn together, they cross at (2, −1).

One solution, none, or infinitely many

Two lines in a plane can meet in only three ways, and slopes tell you which before you draw. Lines with different slopes cross at exactly one point, so the system has one solution. Lines with the same slope and different y-intercepts are parallel and never cross, so there is no solution. Lines with the same slope and the same y-intercept are the same line, which OpenStax calls coincident lines, and every point on it is a solution, so there are infinitely many. The textbook names these cases too: a system with at least one solution is consistent, and one with none is inconsistent.

Graphing has a limit. It shows the answer clearly when the intersection lands on grid points, but if the lines cross at (1/3, 5/7) you cannot read that from a drawing with any confidence. The graph lab below computes the intersection exactly, as a fraction, so you can see how far a reading by eye would be off. Substitution and elimination, in the systems course, find such answers with algebra alone. Graphing is still the best way to understand what a system means.

Systems answer comparison questions. In the linear equations course, two made-up phone plans cost y = 2x + 20 and y = x + 32 dollars for x gigabytes. As a system, the lines meet at (12, 44): at 12 gigabytes both plans cost 44 dollars. To the left of that point the first line is lower, so the first plan is cheaper; to the right the second line is lower. The intersection is the break-even point, and the graph shows at a glance which option wins on each side of it.