Ninth Grade · Math · week 5

Systems: two truths at once

Solve systems of two linear equations by graphing and substitution; interpret the intersection.

One equation with two unknowns has endless solutions — a whole line of them. But TWO equations that must both be true at once? Usually just one point survives: where the lines cross. That crossing point is the system's solution — the only (x, y) loyal to both.

Substitution turns two equations into one: solve either equation for a variable, then substitute that expression into the other. Now it's a one-variable equation — week 2 skills finish the job. Always find BOTH coordinates, and check the pair in BOTH original equations.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Draw both lines of the worked example roughly — SEE the crossing before any algebra.
  • The mantra: "a solution to a system makes BOTH equations true." Test a wrong point to feel it fail one.
  • Substitution choreography: circle the variable you solved for, arrow it into the other equation.
  • Preview the weird cases: parallel lines (no solution), same line (infinitely many) — just as ideas.
Talk about it:
  • Why is the intersection point THE solution?
  • What would it mean if the two lines were parallel?
  • When is substitution easier than graphing?

Mastery looks like: They solve a substitution system, report an ordered pair, and verify it in both equations.

Common stumbles: Finding x and stopping (no y); substituting back into the same equation they came from; losing a negative in 3x = 6-type steps.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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