Lesson 1: Understanding Variables

What are Variables?

Variables are symbols (usually letters) that represent unknown values in mathematical expressions and equations.

Example:

If x represents the number of apples, and you have 5 apples:

x = 5

If you get 3 more apples:

x + 3 = 8

Visual Representation:

Think of variables as empty boxes that can hold different values:

□ + 3 = 8

The box □ represents our variable x

Lesson 2: Solving Linear Equations

Basic Equation Solving

To solve equations, we perform the same operation on both sides to maintain balance.

Example: Solve for x

2x + 5 = 13

Step 1: Subtract 5 from both sides

2x = 8

Step 2: Divide both sides by 2

x = 4

Balance Method Visualization:

Imagine a balanced scale:

Left side: 2x + 5

Right side: 13

Remove 5 from both sides to keep balance!

Lesson 3: Quadratic Equations

Standard Form

Quadratic equations follow the pattern:

ax² + bx + c = 0

Solving by Factoring

Example: Solve x² + 5x + 6 = 0

Step 1: Factor the equation

(x + 2)(x + 3) = 0

Step 2: Set each factor equal to zero

x + 2 = 0 or x + 3 = 0

Step 3: Solve for x

x = -2 or x = -3

Graphical Representation:

The solutions x = -2 and x = -3 are where the parabola crosses the x-axis.

Practice Problems

Problem 1:

Solve for y: 3y - 7 = 14

Problem 2:

Factor and solve: x² - 4x - 12 = 0

Problem 3:

If a number is multiplied by 4 and then 6 is added, the result is 30. Find the number.