Systems of simultaneous equations (internal)
Internal 3 credits 3×3 systems Elimination Unique / no / infinite solutions Planes ModellingAS 91587, 3 credits. Solving three equations in three unknowns — and, more importantly, recognising when a system has one solution, none, or infinitely many, and saying what that means geometrically and in the context it came from.
1 What the grades require
Same ladder for all three Calculus internals.| Grade | What it takes | The half people drop |
|---|---|---|
| Achieved | Select and use methods; know the concepts and terms; communicate with appropriate representations | Showing no working — the method IS the evidence |
| Merit | A logical sequence of steps, connecting concepts, or forming and using a model | The “and also”: relate findings back to the context |
| Excellence | Devise a strategy, build a chain of reasoning or a proof, or form a generalisation | The “and also”: correct mathematical statements, or a stated insight |
Both Merit and Excellence are “one or more of … AND ALSO …”. A perfect answer that never interprets its result in context cannot pass the Achieved ceiling.
2 Setting up a system from a word problem
Define variables explicitly, and check you genuinely have three independent equations.What to do before eliminating anything
- Define EVERY variable in one sentence, with its unit.
- Write one equation per independent piece of given information.
- Check none of the three equations is just a rearrangement of another — that would leave you with only two real constraints.
3 Solving by elimination
Reduce to 2×2, solve, back-substitute — then check against all three originals. ✕Checking against only the equations you used last
A solution can satisfy the last two equations you worked with purely by coincidence, even if an earlier arithmetic slip happened. Always substitute back into ALL THREE original equations as the final step.4 The three outcomes, and how to report them
Elimination is the easy part; the interpretation is the assessed part.| Elimination ends with | Outcome | Geometry | What to write |
|---|---|---|---|
| Values for all three | Unique solution | Planes meet at one point | The point, with units, checked in all three original equations |
| 0 = k, k ≠ 0 | No solution | No common point | The constraints are inconsistent — say which requirement conflicts |
| 0 = 0 | Infinitely many | Planes share a line | A parametric family, then the values that make sense in context |
Always substitute your answer back into all three ORIGINAL equations. It costs a minute and catches the arithmetic slips that otherwise cost the whole question.
5 Writing and interpreting a parametric solution
The family IS the answer — then narrow it using the context. Internal-style A system reduces to x = 2 + 3t, y = 1 − t, z = t, representing kg of three grades Narrow the infinite family to sensible production quantities. Each variable must be non-negative: x ≥ 0 gives t ≥ −2/3; y ≥ 0 gives t ≤ 1; z ≥ 0 gives t ≥ 0. Combined: 0 ≤ t ≤ 1. If quantities must be whole kilograms and t must keep x, y, z all whole, check which values of t in [0, 1] satisfy that — here t = 0 gives (2, 1, 0) and t = 1 gives (5, 0, 1), both valid integer solutions at the endpoints. ✓ The infinite family is real mathematics, but the business only cares about the feasible slice of it — narrowing it to the range that makes physical sense is the "and also" that reaches Merit and beyond.6 Reaching Excellence
A generalisation about the parameter, not a test of individual values.The Excellence move
- Introduce a parameter (often called k) into one coefficient of the system.
- Solve symbolically, in terms of k, to find the CONDITION on k for each of the three outcomes.
- State the result as a generalisation: "the system has a unique solution for all k ≠ 3; at k = 3 it has infinitely many solutions."
Practice
Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.
Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.
Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.
Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.