Algebra
Algebra is the mechanical substrate of Quant — the moment a problem has an unknown, you're doing algebra. This chapter covers the nine algebraic moves that together solve roughly 40% of GMAT Focus Quant questions: isolating variables, handling systems, flipping inequality signs on negatives, the wavy-line method for polynomial and rational inequalities, casework on absolute value, factoring quadratics, substituting into functions and sequences, manipulating symbolic identities, and translating English into equations without losing the word "more" or "less" in the process.
Linear equations in one unknown — isolate-the-variable discipline
The single most important habit in algebra is "isolate the variable before you do anything else." Every one-unknown linear equation reduces to x = something. The work is mechanical; the errors come from doing it sloppily under time pressure.
Mental model. An equation is a balance scale. Whatever you do to one side, you must do to the other — the two sides stay equal. Solving means isolating the unknown by adding, subtracting, multiplying, or dividing both sides at once. Lose this discipline and your work breaks down within three steps.
The four operations that preserve equality:
- Add the same thing to both sides.
- Subtract the same thing from both sides.
- Multiply both sides by the same non-zero thing.
- Divide both sides by the same non-zero thing.
That's it. Every algebraic step reduces to one of these four. If you find yourself doing anything else ("I'll just move this over and forget the sign change"), you're introducing errors.
Example. If 3x + 7 = 22, find 6x + 5.
- Subtract 7:
3x = 15. - Divide by 3:
x = 5. - Compute:
6(5) + 5 = 35.
The faster path: don't always solve for x. Notice 6x + 5 = 2(3x) + 5 = 2(15) + 5 = 35. You only needed 3x, not x. The GMAT rewards students who read what the problem actually asks for — sometimes the target expression is a rearrangement of the given one, and you can skip solving for the individual variable.
The "keep what you need" heuristic. Before solving, glance at what the question wants. If it wants 2x − 1 and your equation gives 2x = 7, just write 2x − 1 = 6. No need to find x.
Clearing fractions and decimals. If an equation has fractions, multiply both sides by the LCD to clear them. x/3 + x/4 = 7 → multiply by 12 → 4x + 3x = 84 → 7x = 84 → x = 12. Decimal coefficients clear the same way: multiply both sides by a power of 10.
Checking by substitution. The cheapest insurance on any algebra problem: plug your answer back into the original equation. If it doesn't satisfy, you made an arithmetic mistake. Ten seconds of checking beats two minutes of confused rework.
Pro tip. Build the substitution check into muscle memory: every time you solve an algebra problem, the last thing you do before bubbling is verify. On the GMAT, an extra 5 seconds per algebra question buys you ~30% fewer arithmetic-slip misses — by far the cheapest accuracy gain available.
Trap to watch. When you multiply or divide both sides by an expression containing a variable, you may be multiplying by zero or flipping a sign. If x − 2 could be negative, multiplying an inequality by x − 2 flips the inequality sign. Keep track of what's in your multiplier.
Self-explanation prompt. In one sentence, why is "isolate the variable" good discipline? If you can say "because once the variable is alone, every remaining step is arithmetic," you've internalized why this is the only reliable path through multi-step algebra.
Systems of equations — substitution vs. elimination
A system of equations has two or more equations and two or more unknowns. On the GMAT, you'll almost always see two equations in two unknowns. Two techniques cover everything.
Substitution — use when one variable is cheap to isolate. If one equation gives you y = 2x + 3, substitute 2x + 3 for y in the other equation. Solve for x, back-substitute for y.
Elimination — use when coefficients line up for addition or subtraction. If one equation has +2y and the other has −2y, adding them kills y.
Example (elimination). x + y = 12 and x − y = 4. Add: 2x = 16, so x = 8. Subtract: 2y = 8, so y = 4. Two lines, no substitution needed.
Example (substitution). At a bakery, 2m + 3s = 21 and 4m + s = 17. From equation 2: s = 17 − 4m. Substitute into equation 1: 2m + 3(17 − 4m) = 21 → 2m + 51 − 12m = 21 → −10m = −30 → m = 3. Then s = 17 − 12 = 5.
Elimination with coefficient matching. If coefficients don't line up, scale one or both equations first. 3x + 2y = 16 and 5x − 2y = 16 line up on y — add them to get 8x = 32, so x = 4. Back-substitute: 3(4) + 2y = 16 → y = 2. Then x + y = 6.
The linear-dependence trap. Two equations in two unknowns don't always pin down the answer. If one equation is a multiple of the other, they describe the same line and the system has infinitely many solutions.
Example (Data Sufficiency). What is x? (1) 2x + 3y = 14. (2) 4x + 6y = 28.
Statement (2) is exactly twice statement (1) — same equation, no new information. The two statements together still form one equation in two unknowns. Answer: E (insufficient together).
Quick test for independence: two linear equations ax + by = c and dx + ey = f are independent if and only if ae − bd ≠ 0. Check this at a glance on Data Sufficiency.
Takeaway. Two equations in two unknowns usually pin down a unique answer — unless one is a linear multiple of the other. Always run the independence check (ae − bd ≠ 0) before claiming sufficiency on Data Sufficiency. The C-trap on this pattern catches a huge fraction of mid-band students.
Trap to watch. Three unknowns generally need three independent equations. But the GMAT can give you three equations where two are redundant, or give you information in a form that's not obviously an equation (a constraint like x > 0). Count independent equations, not visible ones.
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