Graphical Solutions and Number of Solutions

Objective: Identify when two linear graphs meet once.

Prerequisite skills: line equations and solution classification. Two nonvertical lines with different slopes intersect at exactly one point. Solve by setting their y-expressions equal. The x-coordinate solves the equality; the y-coordinate is the shared output. Common error: using the intersection y-coordinate as the solution for x. Restrictions and interpretation: An intersection must lie on both lines; matching slopes alone do not distinguish parallel from identical lines.

Objective: Classify systems by comparing normalized equations.

Write both equations in compatible forms. Equal slopes with different intercepts give no intersection. Equal slopes and equal intercepts describe the same line, so every point on it is a system solution. Common error: calling any equal-slope pair the same line. Restrictions and interpretation: An intersection must lie on both lines; matching slopes alone do not distinguish parallel from identical lines.

Objective: Match all coefficients to obtain infinitely many system solutions.

A multiple of one full equation describes the same line. Multiplying only the variable coefficients without the constant changes the line. Test proportionality across the entire equation. Common error: scaling coefficients but not the constant. Restrictions and interpretation: An intersection must lie on both lines; matching slopes alone do not distinguish parallel from identical lines. Desmos and method choice: In Desmos, graph both complete equations. Different slopes meet once; parallel distinct lines never meet; coincident equations may visually overlap. Verify classification algebraically because a restricted viewing window can hide an intersection.