LibrarySolving Linear Equations

Solving Linear Equations

Learn about Solving Linear Equations as part of GMAT Preparation - Graduate Management Admission Test

Mastering Linear Equations for Competitive Exams

Linear equations are a cornerstone of quantitative reasoning sections in many competitive exams, including the GMAT. Understanding how to solve them efficiently and accurately is crucial for success. This module will break down the fundamentals of solving linear equations, equipping you with the skills needed to tackle these problems with confidence.

What is a Linear Equation?

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable (raised to the power of one). When graphed, a linear equation forms a straight line. The general form of a linear equation with one variable is <code>ax + b = 0</code>, where 'a' and 'b' are constants and 'x' is the variable. For two variables, it's typically represented as <code>ax + by = c</code>.

Solving Linear Equations with One Variable

The goal when solving a linear equation with one variable is to isolate the variable on one side of the equation. This is achieved by applying inverse operations to both sides of the equation, ensuring that the equality remains balanced. The fundamental principle is to perform the same operation on both sides to maintain equilibrium.

What is the first step to solve the equation <code>3x - 7 = 14</code>?

Add 7 to both sides of the equation.

Solving Linear Equations with Two Variables

Linear equations with two variables, such as <code>ax + by = c</code>, have infinitely many solutions. However, in competitive exams, you'll often encounter systems of linear equations, which are sets of two or more linear equations with the same variables. The goal here is to find the specific values of the variables that satisfy ALL equations in the system simultaneously.

MethodDescriptionBest For
SubstitutionSolve one equation for one variable, then substitute that expression into the other equation.Systems where one variable is easily isolated.
EliminationMultiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations together to eliminate that variable.Systems where variables can be easily eliminated by addition or subtraction.

Key Strategies for Competitive Exams

When faced with linear equation problems on timed tests, speed and accuracy are paramount. Practice recognizing common equation structures and applying the most efficient solution method. Always double-check your answers by substituting them back into the original equations.

For GMAT, focus on understanding the underlying logic rather than just memorizing formulas. Many problems can be solved by setting up equations based on the problem's narrative.

Practice Problems and Application

The best way to master solving linear equations is through consistent practice. Work through a variety of problems, starting with simpler ones and gradually moving to more complex scenarios. Pay attention to how different word problems can be translated into algebraic equations.

Visualizing the intersection of two lines on a graph represents the solution to a system of two linear equations. The point where the lines cross is the (x, y) coordinate pair that satisfies both equations simultaneously. This graphical interpretation helps in understanding the concept of a unique solution.

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Learning Resources

Khan Academy: Linear Equations(tutorial)

Comprehensive video lessons and practice exercises covering various aspects of solving linear equations.

GMAT Club: Linear Equations Forum(documentation)

A vast collection of GMAT-specific questions, explanations, and discussions on linear equations.

Magoosh GMAT Blog: Solving Linear Equations(blog)

Tips and strategies specifically tailored for solving linear equations in the GMAT quantitative section.

Math is Fun: Linear Equations(tutorial)

An easy-to-understand explanation of linear equations with examples and interactive elements.

Purplemath: Solving Linear Equations(documentation)

Detailed explanations and step-by-step examples for solving linear equations of various types.

YouTube: GMAT Linear Equations Explained(video)

A video tutorial demonstrating how to solve common GMAT linear equation problems.

Wikipedia: Linear Equation(wikipedia)

A general overview of linear equations, their properties, and applications.

Varsity Tutors: Linear Equation Practice(tutorial)

Practice questions and explanations for linear equations relevant to GMAT preparation.

Brilliant.org: Linear Equations(documentation)

An interactive exploration of linear equations with conceptual explanations and problem-solving techniques.

The Princeton Review: GMAT Math - Linear Equations(blog)

Guidance and practice tips for tackling linear equation questions on the GMAT.