Eleventh Grade · Math · week 5

Systems of linear equations: elimination

Solve systems of two linear equations by elimination, scaling equations when necessary, and interpret the solution in context.

Two linear equations in two unknowns usually cross at exactly one point -- the one (x,y) pair loyal to both. Elimination is the most reliable way to find it: add or subtract the two equations so that one variable cancels out entirely, leaving a single equation in a single variable that Algebra I already taught you to solve.

Sometimes the variables cancel immediately (when their coefficients are already opposites, like +2y and -2y). Other times you must first multiply one or both equations by a constant so a pair of coefficients becomes opposites -- this does not change either equation's truth, since multiplying both sides of an equation by the same number preserves the balance. Once one variable is found, substitute it back into either ORIGINAL equation to find the other, then check the pair in both.

📋 For the teacher
Teach it (10–15 min):
  • Before scaling anything, always check first whether a pair of coefficients is ALREADY opposite (or equal) -- many of these problems need no scaling at all.
  • When scaling is needed, say the plan out loud first: "I will multiply the second equation by -2 so the x-terms become opposites" -- narrating the plan prevents random guessing.
  • Insist on substituting back into an ORIGINAL equation (not a scaled version) to find the second variable, and then checking the final pair in the OTHER original equation.
  • Turn one system into a real story (splitting a bill unevenly, ticket sales, a recipe) so the two variables feel like they mean something, not just letters.
Talk about it:
  • Why does adding two true equations together produce another true equation?
  • How do you decide which variable to eliminate first?
  • Why is it important to check your solution in BOTH original equations, not just one?

Mastery looks like: They solve a two-variable system by elimination, scaling an equation when needed, and check the solution in both original equations without being asked.

Common stumbles: Adding when they should subtract (or the reverse); flipping a sign when scaling an equation; forgetting to find the second variable after finding the first.

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