Algebra System

Algebra is ~35% of Math and almost entirely automatable — translate the words to an equation, then let Desmos or clean elimination finish it. With Advanced Math it makes up ~70% of the section and the bulk of the hard Module 2. For a strong A-Level student these are points to bank flawlessly — the losses here are translation slips and answering the wrong variable, not difficulty.

Sources: College Board math types; College Panda, PWN (plug-in/backsolve); UWorld. Examples MERIDIAN-original. Point convention: each item = 1 of ~44 Math questions; Module-1 misses are ceiling-risk.

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#WHAT IT TESTS

Linear equations in one and two variables; linear functions; systems of two linear equations; linear inequalities (one/two variables, solution sets). Many are word problems ("in context") asking you to build the equation, then interpret a coefficient.

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#THE METHOD

1. TRANSLATE: turn each sentence into an equation. Define variables explicitly (let x = …).
2. SOLVE:
   - one equation → isolate the variable.
   - system → Desmos intersection, OR elimination/substitution.
   - inequality → solve like an equation; FLIP the sign if you ×/÷ by a negative.
3. INTERPRET in context: slope = rate per unit; intercept = starting value.
4. ANSWER THE ASK: re-read — did it want x, or x+y, or "how many more"?

Why (mechanism): the SAT rarely makes the algebra itself hard; it makes the setup and the ask tricky. Defining variables on paper and re-reading the ask removes the two errors that actually cost strong students points. Desmos converts most systems and "meaning/solution-count" items into a 15-second graph.

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#DESMOS PATH (when to graph)

• System: type both equations → click the intersection → exact coordinates. Best for linear-linear and any messy coefficients.
• "How many solutions"/"no solution": add a slider on the unknown constant; drag until lines are parallel (no solution) or coincide (infinite).
• Inequality region: graph each; the shaded overlap is the solution set; test whether a given point lies in it.
• Skip Desmos for trivial one-step solves and for "which expression is equivalent / in terms of a, b" (use algebra or plug-in).

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#TRAP TAXONOMY

• Wrong-variable answer: solved correctly for x, but the question wanted x+y or 2x. (The #1 algebra loss.)
• Sign error in elimination.
• Inequality not flipped when multiplying/dividing by a negative.
• No-solution vs infinite confusion (same slope + different intercept = none; identical = infinite).
• Slope/intercept swap when interpreting meaning in context.

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#SAME ITEM, TWO LEVELS (MERIDIAN-original)

Problem: "A gym charges a £20 joining fee plus £15 per month. A member paid £140 total. The question asks: for how many months, plus the next month's fee, will the member have paid by next month?" Equation: 20 + 15m = 140.

• ~700 (trap): solves 15m = 120 → m = 8, answers 8. Trap: wrong-variable/ask — the question asked for months plus next month's payment context (read the ask), not just m. [−1; in Module 1, a ceiling risk]
• 800 (precise): m = 8 months so far; the ask wants the value it specified — re-read, compute exactly that (e.g., total after next month = 140 + 15 = £155, or "9 months"). Move: isolated m, then re-read the ask and answered the exact quantity. (Annotate the ask before solving — see Careless-Error Elimination.)

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#DIAGNOSE YOUR ANSWER

• Right math, wrong number? You answered the wrong variable. Fix: underline the ask; restate it before solving.
• System wrong? Sign error or arithmetic — switch to Desmos intersection to verify.
• Inequality flipped wrong? You ×/÷ by a negative without flipping. Fix: flag any negative multiply/divide.
• "No solution" missed? Compare slopes and intercepts directly (Formula Vault).

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#DRILLS (timed · self-marked · MERIDIAN-original)

Drill 1 — Translate & answer-the-ask (target 6/6, ≤6 min): for 6 word problems, (a) write "let x = …", (b) write the equation, (c) underline the ask, (d) solve, (e) confirm you answered the ask. PASS: 6/6 with the exact asked quantity. Drill 2 — Desmos systems (target 8/8, ≤8 min): solve 8 systems by graphing only; target <60 s each from reading to clicking the intersection. PASS: all correct; if any wrong, re-enter and find the mis-type. Repetition: fail → repeat next day.

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#TRANSFER

Abstract rule: translate → solve → interpret → answer the ask — for any linear context. Second context: a phone-plan cost problem and a chemistry dilution-rate problem are the same setup (define variable, build equation, answer the ask). Constant: the 4-step method + Desmos paths + trap list. Changes: the story and the numbers.

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#REFERENCE CARD — ALGEBRA

1 Translate (let x = …)  2 Solve (Desmos intersection / elimination / isolate)
3 Interpret (slope = rate, intercept = start)  4 ANSWER THE ASK (x? x+y? how many more?)
Inequality: flip sign on ×/÷ by negative.  #solutions: slopes/intercepts.
TRAPS: wrong variable · sign error · no-flip · none-vs-infinite · slope/intercept swap
Each item = 1 of 44 Math Q; Module-1 miss = ceiling risk.

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MERIDIAN · FORGE · ALGEBRA SYSTEM | v1.0