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

Grade 10 › C10U08 Systems of Linear Equations › L4
GRADE 10 · ALBERTAOUTCOME RF9

L4 · Solving Algebraically: Elimination (Addition)

Learning goal

“I can solve a 2 x 2 linear system by elimination - adding the equations (after scaling one or both, if needed) so one variable cancels - and verify the solution in both original equations.”

Elimination — solving a system by combining the equations so that one variable disappears (its coefficients sum to zero).
Opposite coefficients — coefficients that are negatives of each other, like +2y and -2y; adding the equations cancels that variable.
Scaling an equation — multiplying every term of an equation by the same number, producing an equivalent equation (used to manufacture opposite coefficients).
Equivalent system — a system with the same solution as the original, produced by legal moves (scaling an equation, adding equations).
Worked example

Solve: x + y = 10 and x - y = 4. Step 1: Scan the columns: the y-coefficients are +1 and -1 - opposites. Step 2: Add the equations term by term: (x + x) + (y + (-y)) = 10 + 4 -> 2x = 14. The y variable is eliminated. Step 3: x = 7. Return to either original: 7 + y = 10 -> y = 3. Step 4: Verify in BOTH originals: 7 + 3 = 10 ✓ and 7 - 3 = 4 ✓. Solution (7, 3). Say it plainly: adding two true equations gives a true equation (that is preservation of equality doing the heavy lifting), and when a variable's coefficients are opposites, the addition deletes that variable for free.

Check yourself — explain it

Why does adding x + y = 10 and x - y = 4 eliminate y?

Paired problem — example still visible

Solve 2x + y = 11 and 2x - y = 5, and verify. -> Add: 4x = 16 -> x = 4; 2(4) + y = 11 -> y = 3. Solution (4, 3). Check: 2(4) + 3 = 11 ✓ and 2(4) - 3 = 5 ✓

Faded completion — last step is yours

"3x + 2y = 16 and 3x - 2y = 8: adding gives 6x = ___, so x = 4; then 3(4) + 2y = 16 gives y = ___" -> 24; 2

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