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Properties of Equality

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Properties of Equality

Study Guide

Key Definition

Properties of equality are logical rules that allow you to balance, manipulate, and solve equations: $x = x$, $if\ x = y\ then\ y = x$, $if\ x = y\ and\ y = z\ then\ x = z$

Important Notes

The Reflexive Property states $x = x$
The Symmetric Property states if $x = y$, then $y = x$
The Transitive Property states if $x = y$ and $y = z$, then $x = z$
Addition and Subtraction Properties: if $x = y$, then $x + z = y + z$ and $x - z = y - z$
Multiplication and Division Properties: if $x = y$, then $x \times z = y \times z$ and $\frac{x}{z} = \frac{y}{z}$ when $z \neq 0$

Mathematical Notation

$=$ represents equality

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

Remember to use proper notation when solving problems

Why It Works

These properties allow consistency in solving equations: $x = y$ means applying the same operation to both sides maintains equality.

Remember

Always apply the same operation to both sides of an equation to maintain equality: $x + 3 = y + 3$

Quick Reference

Reflexive Property:$x = x$

Symmetric Property:$x = y \Rightarrow y = x$

Transitive Property:$x = y, y = z \Rightarrow x = z$

Understanding Properties of Equality

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Reflexive Property: For any real number x, x = x. This tells us every quantity is equal to itself.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $x = 5$, what is $x + 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If you have $\$10$ and your friend has the same amount, and you both buy a snack for $\$2$, how much money do each of you have left?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $2x = 10$, what is $x$?

Challenge Quiz

Single Choice Quiz

Advanced

If $x^2 = 16$, what is $x$?

Recap

Watch & Learn

Review key concepts and takeaways