Eighth Grade · Math · week 16

When lines never meet

Recognize systems with no solution (parallel lines) and infinitely many solutions (the same line), connecting to week 3's special cases.

When solving a system by substitution, if the variable cancels: a false statement (like 2 = 8) means NO SOLUTION -- the lines are parallel, same slope but different y-intercepts, and never cross. A true statement (like 0 = 0) means INFINITELY MANY SOLUTIONS -- the two equations describe the exact same line.

A quick preview before solving: same slope, different intercepts predicts no solution; same slope AND same intercept predicts infinitely many; different slopes always predict exactly one solution.

📋 For the teacher
Teach it (10–15 min):
  • A quick slope-check previews the answer before solving: same slope and same intercept means infinite; same slope, different intercept means none; different slopes always mean exactly one.
  • This is the exact same "did the variable cancel, and is what is left true or false" test from week 3, now diagnosing a pair of lines instead of one equation.
  • Ground it physically: parallel train tracks never meet (no solution); the same road with two different names on a map is still one road (infinitely many solutions).
  • Do not let a canceled variable read as "stuck" -- it IS the answer, not a dead end.
Talk about it:
  • If two lines have the same slope but different y-intercepts, will they ever cross? Why?
  • How is a no-solution system different from an infinitely-many-solutions system?
  • Without fully solving, how could you predict a system has exactly one solution just from the slopes?

Mastery looks like: Given a system, they predict from the slopes (or solve and interpret a canceled variable) whether it has no solution, one solution, or infinitely many, with a reason.

Common stumbles: Confusing "no solution" with "the solution equals zero"; not recognizing a system as the same line when it is written in an equivalent but different form.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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