Systems of Equations: Three Methods, One Intersection Point
A system of equations asks where two lines cross. Three ways to find that point — graphing, substitution, elimination — and how to recognize when there are zero, one, or infinite solutions.
Where do two lines meet?
A system of equations is just two equations sharing the same variables. The solution is the point (x, y) that satisfies both equations — visually, the point where the two lines cross.
One linear equation describes infinite points (a whole line). Two linear equations together usually pin down one specific point — the intersection.
Three possible outcomes
What does the solution represent?
A solution to a system of two linear equations is the point where:
Open the question →Infinite solutions
Infinitely many solutions means the two equations represent:
Open the question →Method 1: Substitution
Best when one equation already has a variable isolated (like “y = ...”).
Method 2: Elimination
Best when both equations are in standard form (Ax + By = C). Add or subtract to eliminate one variable.
Method 3: Graphing
Plot both lines and read the intersection. Best for visual confirmation, but coordinates that aren't integers are hard to read off a graph.
3-second recap
- The solution is the point (x, y) that makes both equations true.
- Different slopes → one solution. Same slope, different intercept → no solution. Identical line → infinite.
- One variable already isolated → substitution. Both in standard form → elimination.