GRE Quant · Study & Practice

Linear Equations

AreaAlgebra DifficultyFoundational GRE weightageHigh — the workhorse of GRE Algebra and the setup step behind most word problems

A linear equation is any equation where the variable appears only to the first power — no squares, no roots, nothing exotic. Solving one is a single disciplined idea: do the same thing to both sides until the variable stands alone. That idea covers the plain one-variable solve, rearranging a formula for a different letter, clearing fractions, and untangling the equations hidden inside word problems. The GRE almost never gives you "solve 2x + 3 = 11" nakedly; it wraps the equation in a scenario, or drops it into a Quantitative Comparison where finding the value of x settles which quantity is larger. The basic on-screen calculator handles the final arithmetic, but it cannot rearrange anything — the algebra is on you, and this chapter makes it automatic.

Topics

1 Solving in One Variable 4 worked · 5 practice · 8-Q test Start → 2 Isolating Variables 4 worked · 5 practice · 8-Q test Start → 3 Equations with Fractions 4 worked · 5 practice · 8-Q test Start → 4 Literal Equations 4 worked · 5 practice · 8-Q test Start →

⚡ GRE shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Clear fractions on the first move by multiplying every term by the LCD — you almost never want to add fractions inside an equation.
  • Collect the variable on the side that keeps its coefficient positive; you dodge a whole class of sign errors.
  • If the question asks for an expression like 2x + 1, solve for 2x and stop — you rarely need x itself.
  • Variables cancel to a true statement ⇒ infinitely many solutions; to a false statement ⇒ no solution. Spot this instantly.
  • For a single fraction = fraction, cross-multiply; for a sum of fractions, clear with the LCD instead.
  • Rearranging a formula uses the exact same steps as a numeric solve — treat every other letter as a fixed number.

⚠️ Common mistakes & traps

GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Multiplying only the fraction terms by the LCD and forgetting the whole-number terms.
  • Dividing by the variable (which may be zero) instead of subtracting to collect terms — this can lose solutions.
  • Sign errors when moving a term across the equals sign; each move must flip the sign.
  • Solving all the way to x when the question only wanted 3x or x + 2 — wasting time.
  • When solving a literal equation, dividing by a variable coefficient without noting it must be nonzero.
  • Cross-multiplying an equation that is a sum of fractions, not a single proportion.

📈 GRE exam insight & question patterns

Quantitative Comparison — given 3x − 7 = 11, Quantity A is x and Quantity B is 6. Which is greater?

Solve: 3x = 18, x = 6. The quantities are equal, so the answer is "the two quantities are equal".

Numeric Entry — if (2x + 6)/4 = 5, enter the value of x.

Multiply by 4: 2x + 6 = 20, so 2x = 14 and x = 7.

A formula question gives distance = rate × time and asks you to express rate. What is the answer form?

rate = distance/time. Literal rearranging: divide both sides by time.

Why does 2(x + 3) = 2x + 5 have no solution?

Expanding gives 2x + 6 = 2x + 5; the variables cancel and leave 6 = 5, which is false — so no value of x works.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

ax + b = c ⇒ x =Tap to reveal
(c − b)/a
Clear fractions byTap to reveal
multiplying every term by the LCD
Variables cancel, true statement leftTap to reveal
infinitely many solutions
Variables cancel, false statement leftTap to reveal
no solution
Solve d = rt for tTap to reveal
t = d/r
Solve a/b = c/dTap to reveal
cross-multiply: ad = bc
Fastest when asked for 3x + 1Tap to reveal
find 3x, then add 1 — skip solving for x
Solve P = 2L + 2W for LTap to reveal
L = (P − 2W)/2

📌 Quick revision

Every linear equation yields to one rule — keep both sides balanced while you undo operations in reverse until the variable stands alone. Simplify first, clear fractions with the LCD, collect the variable where its coefficient stays positive, then divide. Watch the two edge cases: variables cancelling to a true statement means infinitely many solutions, to a false one means none. Literal equations are the same moves with letters instead of numbers, and they are the quiet setup step behind word problems and formula questions. On the GRE, often you need an expression rather than x itself — solving for 2x and stopping is faster, and the basic calculator only helps with the final arithmetic, never the rearranging.

Chapter test

📝 Linear Equations — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
📊 My GRE analytics
Track scores, accuracy by topic and progress over time.
View analytics →

🏆 Vidaara GRE success checklist

You have truly mastered Linear Equations when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall8 cards