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

Grade 10 › C10U08 Systems of Linear Equations › L6
GRADE 10 · ALBERTAOUTCOME RF9

L6 · Systems Clinic: 0/1/∞ Verified Algebraically

Learning goal

“I can classify any 2 x 2 linear system TWICE - graphically, by slope and intercept, and algebraically, by what the solving process produces (a unique pair, a false statement like 5 = 8, or an identity like -4 = -4) - and solve the one-solution systems by whichever algebraic method fits.”

Classify — to determine which of the three cases a system falls into: no solution, one solution, or infinitely many solutions.
Contradiction — a false statement with no variables left, like 5 = 8; it is algebra's signature for "no solution."
Identity — a statement true for all values, like -4 = -4, produced when the two equations are the same line; algebra's signature for "infinitely many solutions."
Dependent equations — equations that name the same line (one is a multiple or rearrangement of the other); their system is the infinite-solutions case.
Worked example

Classify 2x + y = 8 and y = -2x + 5, two ways. Graphical way: the second equation IS y = -2x + 5; rewriting the first gives y = -2x + 8. Slopes equal (-2), intercepts differ (8 vs 5): parallel - prediction: no solution. Algebraic way: substitute -2x + 5 for y in the first equation: 2x + (-2x + 5) = 8 -> 5 = 8. The variables vanished and left a FALSE statement. That contradiction is the algebraic signature of no solution - the prediction holds. Say it plainly: assume a common point exists and the algebra prosecutes the assumption. A contradiction means the assumption was impossible; an identity (next) means it was automatic.

Check yourself — explain it

When solving produces 5 = 8, why does that mean NO solution rather than 'I made a mistake'?

Paired problem — example still visible

Classify y = 3x + 2 and 6x - 2y = -4 both ways. -> Graphical: rewrite the second - -2y = -6x - 4, so y = 3x + 2 - identical to the first: coincident, infinitely many. Algebraic: substitute 3x + 2 for y: 6x - 2(3x + 2) = -4 -> -4 = -4: an identity. Both ways agree: infinitely many solutions.

Faded completion — last step is yours

"y = x - 1 and x + y = 5: the slopes are 1 and -1, so there is ___ solution; solving by substitution, x + (x - 1) = 5 gives x = ___ and y = ___." -> exactly one; 3; 2

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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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