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

Grade 10 › C10U08 Systems of Linear Equations › L5
GRADE 10 · ALBERTAOUTCOME RF9

L5 · Systems in Context: Which Method, and What the Solution Means

Learning goal

“I can model a real situation as a system of two linear equations (with a let statement and any domain restrictions), choose an efficient method - graphing, substitution, or elimination - and justify the choice, then state what the solution MEANS in the situation, not just its coordinates.”

Model — a system of equations that represents a situation; the equations are the situation, translated.
Let statement — the sentence that declares what each variable stands for, with units: "Let v = the number of visits."
Break-even point — the solution of a cost-comparison system: the usage level where two options cost exactly the same.
Restriction — a limit the context places on a variable (e.g. gigabytes >= 0, tickets are whole numbers); a mathematical solution outside the restriction is not an answer to the problem.
Worked example

Two gyms, with C the total cost in dollars for v visits: Gym A: C = 30 + 8v (a $30 sign-up fee, then $8 per visit). Gym B: C = 10v + 10 (a $10 sign-up fee, then $10 per visit). Step 1: Both equations are already solved for C, so the efficient move is to set them equal (substitution in its simplest dress): 30 + 8v = 10v + 10. Step 2: Solve: 20 = 2v -> v = 10; then C = 30 + 8(10) = 110. Step 3: Verify: 30 + 8(10) = 110 ✓ and 10(10) + 10 = 110 ✓. Step 4: INTERPRET. (10, 110) means: at 10 visits the gyms cost the same, $110. Under 10 visits, Gym B is cheaper (lower fee); over 10 visits, Gym A is cheaper (lower per-visit rate). The coordinates are not the answer - the recommendation is. Say it plainly: in context, a solution has a first coordinate you can count and a second you can spend. Report both, plus which side of the crossing favours which option.

Check yourself — explain it

Why is reporting just '(10, 110)' an incomplete answer to the gym problem?

Paired problem — example still visible

Two phone plans: Plan P: C = 25 + 3g; Plan Q: C = 15 + 5g, where g is gigabytes of data. Find the break-even point and interpret it. -> 25 + 3g = 15 + 5g -> 10 = 2g -> g = 5; C = 25 + 3(5) = 40. Break-even (5, 40): at 5 GB both plans cost $40; below 5 GB, Plan Q (lower fee) is cheaper; above 5 GB, Plan P (lower per-GB rate) is cheaper. Check: 25 + 15 = 40 ✓ and 15 + 25 = 40 ✓

Faded completion — last step is yours

"Taxi X: C = 4 + 2.5d; Taxi Y: C = 2 + 3d (d in km). Setting them equal: 4 + 2.5d = 2 + 3d gives 2 = ___d, so the trips tie at d = ___ km, costing $___." -> 0.5; 4; 14

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