Predict before solving
Line geometry and coefficient ratios tell whether a pair has one, none or infinitely many solutions before lengthy solving.
Learning goals
- Connect line relationships to solution counts and consistency terms.
- Apply coefficient-ratio criteria after standardising both equations.
- Determine parameters producing parallel or coincident lines.
Concept lesson
Three line relationships
Intersecting lines: one solution. Parallel distinct lines: no solution. Coincident lines: infinitely many solutions.
Consistency vocabulary
A pair is consistent if it has at least one solution. One solution means consistent and independent; infinitely many means consistent and dependent; no solution means inconsistent.
Ratio decision table
For a1x+b1y+c1=0 and a2x+b2y+c2=0: unequal a and b ratios imply one solution; equal a and b ratios but unequal c ratio imply no solution; all three equal imply infinitely many. Standardise signs first.
Use all required ratios
For x+2y-4=0 and 2x+4y-12=0, a1/a2=1/2 and b1/b2=1/2, but c1/c2=1/3. Therefore the lines are parallel and the pair has no solution.
Classify a coincident pair
For 2x+3y-9=0 and 4x+6y-18=0, all ratios equal 1/2. The same line is represented twice: infinitely many solutions, consistent and dependent.
Find a coincidence parameter
For x+2y=3 and 2x+4y=k to be coincident, write both in standard form. Since the second left side is twice the first, k must equal 6.
Complete the decision table
For three generated pairs, record ratios, line relationship, solution count, consistency and dependence. The checker requires every column to agree.
Explain an algebraic contradiction
If elimination produces 0=5, explain why no ordered pair can satisfy both equations and connect this to parallel lines and inconsistency.
One-mark consistency pattern
Standardise, compare ratios in the correct order, then state both solution count and line relationship when asked.
Exit ticket
Classify 2x+3y=7 with 4x+6y=14 and with 4x+6y=10. Then find k so that 3x+ky=5 is parallel to 6x+4y=9.