Math 1 · Unit 6 · Lesson 4
How Many Solutions?
Two lines cross once, never, or everywhere. Different slopes give one solution. Parallel lines (same slope, different y-intercept) give none, and the algebra ends in a false line like 0 = 5. The same line twice gives infinitely many, and the algebra ends in a true line like 0 = 0. Compare m and b and you can call it before solving.
The big idea
Two lines can cross once, never, or everywhere, so a system of two linear equations has one solution, no solution, or infinitely many. The picture and the algebra tell the same story. Different slopes cross once, and the algebra gives x a single value. Parallel lines, same slope and different y-intercept as in Unit 4, never meet, and the algebra ends in a false line like 0 = 5 because the x terms cancel and the leftover intercepts disagree. The same line written twice meets everywhere, and the algebra ends in a true line like 0 = 0. Put both equations in y = mx + b, compare, and you can call the number of solutions before you solve anything.
The parts of this lesson
Each part is its own short page. Go in order, or pick the one you need.
Printables
One page with the three pictures, the m and b prediction, what a false line and a true line mean, and a worked example of each. Plus a worksheet with grids and an answer key, and a short lesson plan.