Graphical Meaning & Special Cases of Systems
Flashcards on interpreting intersections, parallel lines, identical lines, and 'find the constant' PSAT questions.
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Questions Covered in This Set
10 cards to master
What does a solution to a system represent graphically?
A point that lies on both lines — an intersection point.
What are the three possible numbers of solutions for two lines?
Exactly one (lines cross), none (parallel), or infinitely many (same line).
Slopes and intercepts for NO solution
Same slope, different y-intercepts — parallel lines.
Slopes and intercepts for INFINITELY many solutions
Same slope AND same y-intercept — identical lines.
Fast algorithm for counting solutions
Put both equations in y = mx + b; compare slopes first, then compare intercepts.
Elimination ends in 0 = 8. What does that mean?
A false statement — the system has no solution (parallel lines).
Elimination ends in 0 = 0. What does that mean?
A true statement — infinitely many solutions (same line).
Proportion condition for parallel-or-identical lines in ax + by = c form
a₁/a₂ = b₁/b₂ (coefficients proportional).
Extra condition needed for infinitely many solutions
a₁/a₂ = b₁/b₂ = c₁/c₂ — the constants must match too.
In a word problem about two pricing plans, what does the intersection mean?
The point where both plans cost the same amount.
