How to apply properties of equality to solve basic equations

📚 Understanding Properties of Equality

Properties of equality are fundamental rules that allow us to manipulate equations while maintaining their balance. Think of an equation like a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. These properties are the bedrock of solving algebraic equations.

📜 A Brief History

The concept of equality and its properties has been around since the development of algebra. Early mathematicians in ancient civilizations, such as the Babylonians and Egyptians, used these ideas implicitly when solving practical problems. The formalization of these properties came later with the development of symbolic algebra.

➗ Key Principles of Equality

• ➕ Addition Property: If $a = b$, then $a + c = b + c$. Adding the same value to both sides doesn't change the equality.
• ➖ Subtraction Property: If $a = b$, then $a - c = b - c$. Subtracting the same value from both sides maintains the equality.
• ✖️ Multiplication Property: If $a = b$, then $a \* c = b \* c$. Multiplying both sides by the same value keeps the equation balanced.
• ➗ Division Property: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$. Dividing both sides by the same non-zero value preserves the equality.
• 🔁 Reflexive Property: $a = a$. Any value is equal to itself.
• ↔️ Symmetric Property: If $a = b$, then $b = a$. The equation can be read from left to right or right to left.
• 🔗 Transitive Property: If $a = b$ and $b = c$, then $a = c$. If two values are equal to the same value, they are equal to each other.
• 💡 Substitution Property: If $a = b$, then $a$ can be substituted for $b$ in any expression.

🧮 Real-World Examples

Let's look at some examples of how to apply these properties to solve equations:

Example 1:

Solve for $x$: $x + 5 = 12$

1. Apply the subtraction property: Subtract 5 from both sides: $x + 5 - 5 = 12 - 5$
2. Simplify: $x = 7$

Example 2:

Solve for $y$: $3y = 15$

1. Apply the division property: Divide both sides by 3: $\frac{3y}{3} = \frac{15}{3}$
2. Simplify: $y = 5$

Example 3:

Solve for $z$: $z - 8 = 4$

1. Apply the addition property: Add 8 to both sides: $z - 8 + 8 = 4 + 8$
2. Simplify: $z = 12$

Example 4:

Solve for $a$: $\frac{a}{2} = 6$

1. Apply the multiplication property: Multiply both sides by 2: $2 \* \frac{a}{2} = 2 \* 6$
2. Simplify: $a = 12$

📝 Conclusion

The properties of equality are indispensable tools for solving equations. By understanding and applying these properties, you can confidently manipulate equations to find the values of unknown variables. Remember to always maintain balance by performing the same operation on both sides of the equation. Happy solving!