Lesson 2.3.2

Writing Algebraic Proofs

Solve an equation step by step, and justify every step— that's an algebraic proof. It's the training ground before geometric proofs.

Introduction

You already know how to solve equations. An algebraic proof simply adds reasons to every step, using the properties from the previous lesson. Think of it as showing your work — with labels.

Past Knowledge

Properties of Equality (2.3.1). Solving linear equations.

Today's Goal

Write a two-column algebraic proof with statements and reasons for each step.

Future Success

Two-column geometric proofs (2.3.3) use the exact same format.

Key Concepts

Two-Column Format

StatementsReasons
What you know or deriveWhy it's true (property, given, definition)

Common Reason Categories

  • Given — the starting information
  • Properties of Equality — addition, subtraction, multiplication, division, substitution
  • Simplify / Combine like terms — arithmetic manipulation

Always Start with “Given”

The first statement in a proof is always the given information, and the reason is simply “Given.”

Worked Examples

Complete Proof

Solving a Linear Equation

Given: . Prove: .

StatementsReasons
Given
Subtraction Property of Equality
Division Property of Equality
Intermediate

Multi-Step Proof

Given: . Prove: .

StatementsReasons
Given
Multiplication Property of Equality
Addition Property of Equality
Advanced

Distributive Property Proof

Given: . Prove: .

StatementsReasons
Given
Distributive Property
Simplify (combine like terms)
Subtraction Property of Equality
Division Property of Equality

Common Pitfalls

Skipping Steps

In a proof, every algebraic manipulation needs its own line and reason. Don't combine multiple operations into one step.

Real-Life Applications

Audit Trails & Accounting

Accountants justify every adjustment to a ledger with a reason (receipt, invoice, regulation). An algebraic proof is the mathematical version of an audit trail — every step is documented and verifiable.

Practice Quiz

Loading...