Variables on both sides - Algebra-1

1. Fundamental Concepts

Definition: Equations with variables on both sides are algebraic equations where the variable appears on both the left and right sides of the equation.
Objective: The goal is to isolate the variable on one side of the equation.
Steps: Typically involves combining like terms, using inverse operations, and simplifying expressions.

2. Key Concepts

Basic Rule: $$4x + 3 = 2x - 5$$

Degree Preservation: The highest degree in the result matches input

Application: Used to solve real-world problems involving multiple unknowns

3. Examples

Example 1 (Basic)

Problem: Solve $$3x + 4 = 2x - 7$$

Step-by-Step Solution:

Subtract $2x$ from both sides: $$3x - 2x + 4 = -7$$
Simplify: $$x + 4 = -7$$
Subtract 4 from both sides: $$x = -11$$

Validation: Substitute $x = -11$: Original: $3(-11) + 4 = 2(-11) - 7 \rightarrow -33 + 4 = -22 - 7 \rightarrow -29 = -29$ ✓

Example 2 (Intermediate)

Problem: $$5y - 3 = 2y + 8$$

Step-by-Step Solution:

Subtract $2y$ from both sides: $$5y - 2y - 3 = 8$$
Simplify: $$3y - 3 = 8$$
Add 3 to both sides: $$3y = 11$$
Divide by 3: $$y = \frac{11}{3}$$

Validation: Substitute $y = \frac{11}{3}$: Original: $5(\frac{11}{3}) - 3 = 2(\frac{11}{3}) + 8 \rightarrow \frac{55}{3} - 3 = \frac{22}{3} + 8 \rightarrow \frac{55}{3} - \frac{9}{3} = \frac{22}{3} + \frac{24}{3} \rightarrow \frac{46}{3} = \frac{46}{3}$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use a balance scale analogy to visualize moving terms across the equal sign.
Error-Proofing: Double-check each step by substituting the solution back into the original equation.
Concept Reinforcement: Practice with a variety of problems that involve different types of coefficients and constants.