Grade 10 — Mathematics 10C › C10U08 Systems of Linear Equations

Grade 10 › C10U08 Systems of Linear Equations › L2
GRADE 10 · ALBERTAOUTCOME RF9

L2 · Zero, One, or Infinitely Many Solutions (Graphical View)

Learning goal

“I can look at the two lines of a system - by their slopes and y-intercepts - and predict whether the system has no solution, exactly one solution, or infinitely many solutions, and I can justify the prediction from the graph.”

Parallel lines — lines with the same slope that never meet; in a system they produce no solution.
No solution — the classification of a system whose graphs never intersect; no ordered pair satisfies both equations.
Infinitely many solutions — the classification of a system whose two equations graph as the SAME line; every point on the line is a solution.
Coincident lines — two equations whose graphs lie exactly on top of each other (one line drawn twice).
Worked example

Consider y = 2x + 3 and y = 2x - 5. Step 1: Compare slopes: both are 2 - the lines tilt identically. Step 2: Compare y-intercepts: 3 versus -5 - the lines are shifted apart. Step 3: Lines with the same slope and different intercepts are parallel: they never meet, so there is no point on both, and the system has no solution. Say it plainly: a solution of a system is a shared point. Parallel lines share nothing - not "the solution is far away," not "the solution is hard to find": there isn't one.

Check yourself — explain it

Why do equal slopes with different y-intercepts guarantee no solution?

Paired problem — example still visible

Classify y = -x + 4 and y = -x - 2: how many solutions? -> No solution: both slopes are -1, intercepts 4 and -2 differ, so the lines are parallel.

Faded completion — last step is yours

"y = 3x + 1 and y = 3x + 1 have the same ___ and the same ___, so their graphs are the ___ line, and the system has ___ solutions." -> slope; y-intercept; same (coincident); infinitely many

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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