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)

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$$ 15 $$
Interactive Practice

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