Equation Solving Principles

Order of Operations

The order of operations is a fundamental principle in mathematics that determines the sequence in which calculations are performed. Understanding and applying the correct order ensures accurate results when solving equations.

• Parentheses (Grouping): Always evaluate expressions inside parentheses first.

• Exponents (Powers): Next, calculate powers such as squares and cubes.

• Multiplication & Division: Perform multiplication and division from left to right.

• Addition & Subtraction: Finally, carry out addition and subtraction from left to right, unless they are inside parentheses.

Example:

• Evaluate Step 1: Multiplication: Step 2: Addition: Final Answer: $14$

• Evaluate Step 1: Parentheses: Step 2: Division: Final Answer: $7$

Mixed Sign Arithmetic

Calculations often involve both positive and negative numbers. Understanding how to combine these is essential for accurate computation.

• Positive + Positive: The result is more positive.

• Negative + Negative: The result is more negative.

• Positive + Negative: The sign of the larger absolute value determines the result; subtract the smaller from the larger.

Examples:

Double Negation

Double negation refers to the property that the negative of a negative number is positive.

• Rule:

Examples:

Multiplication of Negatives

Multiplying two negative numbers results in a positive number.

• Rule:

Example:

Distribution (Distributive Property)

The distributive property allows multiplication to be distributed over addition or subtraction within parentheses.

• Rule:

Examples:

Solving Equations

Definition and Solution

An equation is a mathematical statement that asserts the equality of two expressions, often involving an unknown variable. The solution to an equation is the value of the variable that makes the equation true.

• Example: Step 1: Subtract 3 from both sides: Step 2: Divide both sides by 2: Verification: Substitute into the original equation:

Equation Solving Properties (Steps)

Solving equations involves a systematic approach to isolate the variable and verify the solution.

• Balance: Apply the same arithmetic operation to both sides of the equation.

• Cancelation: Use opposite operations to eliminate terms (e.g., subtract to cancel addition).

• Isolation: Rearrange the equation to get the variable alone on one side.

• Verification: Substitute the solution back into the original equation to check correctness.

Examples:

• Solve Step 1: Add 7 to both sides: Step 2: Divide both sides by -5: Verification:

• Solve Step 1: Expand both sides: Result: Step 2: Move all terms with to one side and constants to the other: Step 3: Divide both sides by -30: Verification: Substitute into the original equation.

Recap Lesson

Order of Operations Summary

• i) Parentheses

• ii) Exponents

• iii) Multiplication & Division

• iv) Addition & Subtraction

Key Arithmetic Rules

• Negative plus Negative is more negative.

• Negative times Negative is positive.

Equation Solving Steps Summary

• i) Balance

• ii) Cancelation

• iii) Isolation

• iv) Verification

Practice Exercises

For further practice, refer to textbook exercises:

• J.5: 1, 3, 5, 7, 11, 13, 15

• J.9: 1, 2, 3, 8

• J.17: 3, 4, 5, 6, 7, 8