Equations Are Lines: Solutions Are Crossings
Each linear equation in two unknowns is a line, and a solution of the system is where the lines cross. So 'no solution' is parallel lines, 'infinitely many' is the same line, an equation saying a product of two linear expressions is zero is the union of two lines, and three distinct lines fail to make a triangle exactly when two are parallel or all three pass through one point.
grade 8 ExpressionsFunctions Draw a DiagramEliminate Possibilities
Progression (7)
g8 1. Find the common point from the known lines Expressions
g8 2. A zero product graphs as two lines Expressions
g8 3. A second solution forces the two lines to coincide Expressions
g8 4. Two lines through the origin meeting twice coincide Expressions
g8 5. Swapped-coefficient lines always meet at x = 1 ExpressionsNumber system
g8 6. When three lines fail to make a triangle ExpressionsFunctions
g8 7. No triangle: a parallel pair or one common point ExpressionsFunctions