GRE Linear Equations: A Guide to Efficient Problem Solving

Linear equations are a cornerstone of the GRE Quant section. While the on-screen calculator can help, true speed comes from mastering the algebraic methods. This guided path will teach you to translate and solve complex word problems efficiently.

Section 1: The Core Algebraic Engine

Building the Foundation: Solving for a Variable

Start by reinforcing the fundamental skill of isolating a variable. Our interactive balancer visually demonstrates the step-by-step process of solving any linear equation.

Step 1The Equation

$$ 3x + 5 = 17 $$

Goal: Isolate x.

Step 2Step 1: Subtract 5

Subtract 5 from both sides to remove the constant.

$$ 3x = 17 - 5 $$
$$ 3x = 12 $$

Step 3Step 2: Divide by 3

Divide both sides by 3 to isolate x.

$$ x = \frac{12}{3} $$
$$ x = 4 $$

Core Methods: Solving Systems of Equations

Learn the two essential algebraic methods—Substitution and Elimination—which are often faster than using the calculator for finding a unique solution.

Step 1The System

$$ x + y = 10 $$
$$ x - y = 2 $$

Step 21. Isolate Variable

From Eq 1: $$ x = 10 - y $$

Step 32. Substitute

Plug into Eq 2:
$$ (10 - y) - y = 2 $$
$$ 10 - 2y = 2 $$

Step 43. Solve

$$ 8 = 2y \implies y = 4 $$
$$ x = 10 - 4 = 6 $$
Solution: (6, 4)
1Tab 1 of 2•Substitution Method

Section 2: GRE-Specific Applications

The Art of Translation: Building Equations

The GRE is famous for its word problems. This interactive guide teaches the crucial first step: deconstructing a problem statement and translating it into a solvable equation.

Interactive Sentence Translator

A number is doubled and then increased by 7. The result is 25.
Equation Build-Up
$$ 2x $$

Application: Tackling Complex Scenarios

Move on to more complex GRE-style scenarios (like age or mixture problems) that require you to build and solve a complete system of two equations.

Systems of Equations: Word Problems

The sum of the present ages of a father and his son is 60 years. Six years ago, the father's age was five times the age of the son. What is the son's present age?
Equation Build-Up
$$ F + S = 60 $$

Strategy: Quantitative Comparison with Equations

Learn how to quickly compare two quantities involving linear equations without necessarily solving for a final value.

Given: $$ 4x + 6y = 10 $$

Quantity A$$ 6x + 9y $$
Quantity B$$ 15 $$
1Tab 1 of 3•The Problem
Interactive Practice Series2 Timed Drills Available

Ready to test your mastery of Linear Equations?

Put the concepts and shortcuts you just learned to the test. Choose from our curated timed practice tests below:

10 Qs • 20m

Linear Equations Practice Test 1

Problem Solving & Data Sufficiency mixed timed drill.

10 Qs • 20m

Linear Equations Practice Test 2

Problem Solving & Data Sufficiency mixed timed drill.