Introduction to Linear Equations

Introduction

A linear equation is one of the most fundamental tools in algebra. It represents a straight-line relationship between variables and appears everywhere — from calculating costs to understanding rates of change. In this lesson, we'll build a rock-solid understanding of what linear equations are and exactly how to solve them.

What Is a Linear Equation?

A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a ≠ 0. The term "linear" refers to the fact that the variable x appears only to the first power — no x², no √x, just plain x.

ax + b = c

Think of an equation as a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.

The Golden Rules of Equation Solving

To solve a linear equation, we isolate the variable by performing the same operations on both sides. The four properties we use are: Addition Property (add the same value to both sides), Subtraction Property, Multiplication Property, and Division Property.

Solving Multi-Step Equations

When equations have multiple terms, we follow a systematic process: (1) Distribute if needed, (2) Combine like terms on each side, (3) Move variable terms to one side, (4) Move constant terms to the other side, (5) Isolate the variable by dividing.

2(x + 3) - 4 = 3x - 1

Key Terms Glossary

Linear Equation

An equation where the variable appears to the first power only, forming a straight line when graphed.

Solution

The value of the variable that makes the equation true.

Isolate

To get the variable alone on one side of the equation.

Coefficient

The numerical factor multiplied by the variable (e.g., in 3x, the coefficient is 3).

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