Systems of equations (internal)
Internal 2 credits Linear + non-linear Inequations Feasible region Simultaneous equations ModellingAS 91269, 2 credits. This standard is specifically about a pair of simultaneous equations where ONE IS NON-LINEAR (e.g. a line and a parabola), AND systems of linear INEQUATIONS — not pure linear elimination. Interpreting what a solution actually means in context is where the grade sits.
1 Reading the notation
A linear equation paired with a non-linear one, and systems of linear inequations — read this before anything else, since it corrects a common assumption about what this standard covers.📖 Terms the schedule grades you on
4 terms- Apply methods (Achieved)
- Select and use methods; demonstrate knowledge of concepts and terms; communicate using appropriate representations.
- Named methods (note 4)
- A pair of simultaneous equations, one non-linear; a system of linear inequations; connecting representations; interpreting solutions in context.
- Non-linear
- Not a straight line — most commonly a quadratic (parabola) or a circle, paired with one linear equation.
- Inequation
- A linear relationship using <, >, ≤ or ≥ instead of =, graphed as a shaded region rather than a line.
2 What the grades require
The same ladder every L2 maths internal uses.| 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.
3 A linear equation paired with a non-linear one
Substitute, solve the resulting quadratic, and connect the count of solutions to the geometry. Internal-style Solve y = x + 1 and y = x² − x − 1 simultaneously How many solutions, and what do they mean? Substitute: x² − x − 1 = x + 1. Rearrange: x² − 2x − 2 = 0. Discriminant: (−2)² − 4(1)(−2) = 4 + 8 = 12 > 0 → TWO real solutions. Solve: x = (2 ± √12)/2 = 1 ± √3, then substitute back into y = x + 1 for each. GEOMETRICALLY: the line crosses the parabola at two distinct points — connecting the algebra (discriminant) to the picture (two intersections). ✓ Two solutions, because the discriminant is positive — the line genuinely crosses the curve twice.4 Systems of linear inequations
Boundary lines, shaded regions, and the feasible region where every constraint overlaps.Graphing a system of inequations
- Draw each boundary line — solid for ≤/≥, dashed for </>.
- Test a point (usually the origin) in each ORIGINAL inequation to decide which side to shade.
- The feasible region is where every shaded region overlaps.
- Vertices of that region come from solving pairs of boundary equations simultaneously.
5 Forming, checking, and interpreting
The context step is where a correct system stops being just algebra. ✕A correct solution with no interpretation
Finding x = 12, y = 8 and stopping there sits at Achieved at best. The standard specifically names "interpreting solutions... in context" as a method area — say what the numbers mean, in the units of the actual problem, and rule out any solution that does not make physical sense.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.