Intersections & Solution Types
Interpret a system as several constraints that must be satisfied at the same time, then distinguish one shared solution, no shared solution, and infinitely many shared solutions.
What does it mean to solve two equations at the same time?
Each equation allows its own set of ordered pairs. A system keeps only the pairs that survive every constraint. Geometrically, that means finding the intersection of the solution sets.
Graphing, substitution, and elimination will all search for this same shared set in different ways.
One point must make both equations true.
For y = x + 2 and y = −x + 4, the lines cross at (1, 3). Substitution into both equations confirms why the intersection is the system solution.
(1, 3) satisfies y = x + 2.
The same point also satisfies y = −x + 4.
Because both constraints are true, (1, 3) solves the system.
Distinct nonparallel lines cross once.
Distinct parallel lines never meet.
Equivalent equations share every point.
Read the geometry, then name the shared solution set.
For the one-solution model, compare a shared point with a point that satisfies only one equation.