Linear Equations

Definition

A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$. It is a first-degree equation because the highest power of the variable is one.

• How to read: “The equation a x plus b equals zero, where a is not zero.”
• Meaning: Simplest equation in one unknown; isolating $x$ gives exactly one solution.

Why It Matters

Linear equations are the simplest “balance sheets” of logic. They provide the foundation for all quantitative reasoning.

Core Concepts

• Solution/Root: The value that makes the equation a true statement. A linear equation has exactly one solution: $x = -\frac{b}{a}$.

• How to read: “The value x equals negative b divided by a.”
• Meaning / when to use: Closed-form solution after moving $b$ to the other side and dividing by $a$.

• Equivalent Equations: Equations that have the same solution set. Operations that produce equivalent equations include adding/subtracting the same value from both sides or multiplying/dividing by a non-zero constant.
• Solving Strategy:
  1. Clear fractions by multiplying by the Least Common Multiple (LCM) of denominators.
  2. Simplify both sides (distribute and combine like terms).
  3. Isolate the variable term on one side.

Connected Concepts

• Linear Functions
• Problem Solving in Mathematics