Solve One-step equations - Part 2

Math 6

1. Fundamental Concepts

Definition: One-step equations are equations that can be solved in a single step by performing the same operation on both sides of the equation.
Inverse Operations: Addition and subtraction are inverse operations, as are multiplication and division.
Isolation of Variable: The goal is to isolate the variable on one side of the equation to find its value.

2. Key Concepts

Addition/Subtraction Property of Equality: $$a + c = b + c \quad \text{and} \quad a - c = b - c$$

Multiplication/Division Property of Equality: $$a \cdot c = b \cdot c \quad \text{and} \quad \frac{a}{c} = \frac{b}{c} \quad (c \neq 0)$$

Application: Used to solve real-world problems involving simple linear relationships

3. Examples

Example 1 (Basic)

Problem: Solve for $$ x $$ in the equation $$ x + 5 = 12 $$.

Step-by-Step Solution:

Subtract 5 from both sides: $$ x + 5 - 5 = 12 - 5 $$
Simplify: $$ x = 7 $$

Validation: Substitute $$ x = 7 $$ into the original equation: $$ 7 + 5 = 12 $$ ✓

Example 2 (Intermediate)

Problem: Solve for $$ y $$ in the equation $$ 3y = 18 $$.

Step-by-Step Solution:

Divide both sides by 3: $$ \frac{3y}{3} = \frac{18}{3} $$
Simplify: $$ y = 6 $$

Validation: Substitute $$ y = 6 $$ into the original equation: $$ 3 \cdot 6 = 18 $$ ✓

4. Problem-Solving Techniques

Identify the Operation: Determine whether the equation involves addition, subtraction, multiplication, or division.
Use Inverse Operations: Apply the inverse operation to both sides of the equation to isolate the variable.
Check Your Solution: Substitute the solution back into the original equation to verify it is correct.