Pure Mathematics 1
Topic Brief: Simultaneous Equations By Substitution
SIGNAL Topic Master Brief: Simultaneous Equations by Substitution (WMA11 §1.6)
Provenance — written 2026-09-29. Built from research/veridian/WMA11-verified-facts.md (grepped for
"simultaneous" and "Jan 2019 Q11", read in full at §4.1, §6, §7 and the lesson-audit log); the existing
content/veridian/curriculum/wma11/simultaneous-equations-by-substitution.ts lesson header (read in full,
not re-taught here); the three companion technique-hub documents already in this folder —
codex-deep-dive-simultaneous-equations-by-substitution.md, forge-exam-technique-guide.md §4.1, and
signal-frequency-and-boundaries.md §1 — and content/veridian/question-bank/wma11.ts. No new primary PDF
was fetched; every citation below is one already verified in those sources. This brief adds three things
none of those documents state outright: a synthesized mark-tariff pattern, the "disguised topic" framing of
where this skill actually turns up in a paper, and a short expect-this-if-it-appears summary. It does not
re-teach the mechanism (CODEX §2), re-list the sub-cases (CODEX §3), or repeat the trap-taxonomy prose
(the lesson file, FORGE §4.1) — see those documents for that.
Every real series found testing this topic
| Series / question | Marks | What was actually tested |
|---|---|---|
| Jan 2019, Q11(b)–(c) | 3 + 5 | "Show that" the x-coordinates of two intersecting curves (a quadratic and a cubic, no line involved) satisfy a given cubic equation, then find the first-quadrant point exactly (surds). |
| Jan 2023, Q7(c) | 2 | Substitute a known coordinate directly into to find — a line meeting the reciprocal curve . |
| Jan 2023, Q7(d) | 4 | Use that to find the second intersection point: cross-multiply the reciprocal into a solvable quadratic, then convert the surviving back to . |
| Jan 2023, Q2(b) | 2 (of 4) | Two linear equations solved simultaneously, as one of three valid routes to a rectangle's fourth vertex — a coordinate-geometry question, not a labelled "solve these equations" question. |
| June 2023, Q2 | 6 | A perimeter-and-area word problem translated into two equations, solved to a 3-term quadratic, then one of two valid root-pairings rejected using a stated constraint. |
That is four real series with a confirmed spec-1.6-by-substitution appearance: Jan 2019, Jan 2023 (twice, in two different questions), and June 2023.
Frequency, stated honestly
WMA11-verified-facts.md treats five series as fully read for a genuine examiner-report mine (Jan 2019,
Oct 2020, Oct 2021, Jan 2023, Jan 2025), plus two more where only one targeted question was checked (June
2022 Q5, June 2023 Q1–Q2) rather than the whole paper. Two further series (Oct 2022, Jan 2024) were
confirmed as genuine Pearson documents but never mined question-by-question, so they cannot be scored
either way for this topic. Against that base:
- Seen in 2 of 5 series checked in full (Jan 2019, Jan 2023) — this is a broader count than
signal-frequency-and-boundaries.md's own table, which reports "1 of 5" because it scores only the narrowest shape (a line meeting a non-linear curve, i.e. Jan 2023 Q7 alone) and doesn't fold in Jan 2019 Q11's curve-meets-curve pairing, which carries no line at all but is still genuine spec-1.6-by-substitution content per the facts file's own §4.1 entry. Both counts are defensible depending on how narrowly "this topic" is drawn; neither is a majority. - Seen in 1 additional series checked by a single targeted question (June 2023 Q2) — June 2022's one targeted question (Q5) turned out to be a different spec point (reconstructing a quadratic from its turning point), not this topic, so June 2022 does not count as a sighting.
- Combining every series that received any check at all (5 full + 2 targeted = 7), the topic was seen in 3 of 7 series checked.
- Oct 2022, Jan 2024, and every June series other than 2022/2023 remain unchecked for this specific topic — not absence of evidence, just absence of a check. No percentage or likelihood claim is made anywhere in this brief; only these plain counts.
The real mark-tariff pattern
In every confirmed sighting, this topic is never a standalone 1–2 mark question — it appears as one part of a multi-part question, worth 4–8 marks in total across two or more sub-parts, and the marks are coded in a consistent shape: an M1 for the algebraic move that sets up or eliminates a variable (substituting a known value, or forming the two equations from prose), a dependent mark (dM1) for solving whatever equation that move produced, and one or two A1s for the resulting value(s) — with a final A1 sometimes gated on a step after the algebra (converting back to in Jan 2023 Q7(d); assigning and to the correct variable under a stated constraint in June 2023 Q2). The heaviest single sighting is June 2023 Q2 at 6 marks across a B1/B1/M1/dM1/A1/A1 chain; the lightest standalone piece is Jan 2023 Q7(c) at 2 marks, but it never appears alone — it is always paired with a heavier follow-on part.
The recurring context/angle: this topic is usually disguised
Only one of the four real sightings (Jan 2023 Q7) is a question that announces itself as "solve these simultaneous equations." The other three bury the same technique inside a different headline task: a coordinate-geometry construction (find a missing vertex — Jan 2023 Q2b, where the mark scheme's own alternative methods make substitution the least reliable of three routes in practice), a "show that" verification of two curves' intersection (Jan 2019 Q11), and a real-world modelling problem stated entirely in prose (June 2023 Q2). The common thread across all four is that solving the equations is a means to a different stated goal — a vertex, a proof, a physical quantity — never the question's whole point by itself.
If this topic appears, here's what to expect
Expect it hidden inside a longer, differently-titled question rather than announced by name. Expect the credited method to reward substituting a known value immediately rather than eliminating symbolically first (Jan 2023 Q7(c)'s examiner report confirms the symbolic route caused more algebra errors). Expect a final step after the equation is solved — converting back to the other variable, or choosing between two valid answers via a constraint stated only in the question's prose — to be where marks are actually lost. Expect the second equation to be almost anything (reciprocal curve, second linear equation, a second non-linear curve, or two equations built from a word problem) rather than the quadratic that most textbook treatments (including this course's own worked examples) default to for teaching the mechanism.