# Linear Equations

**Linear Equations**

Linear equations are fundamental in mathematics and find applications in various fields. They describe relationships that are proportional and can be represented as:

**ax + b = 0**

Where:

**x**is the variable you want to solve for.**a**and**b**are constants.

**1. Types of Linear Equations:**

a. **One-Variable Linear Equations:** - These equations involve a single variable, typically denoted as **x**. - Example: 2x + 5 = 11

b. **Two-Variable Linear Equations:** - These equations involve two variables, often **x** and **y**. - Example: 3x + 2y = 8

c. **Systems of Linear Equations:** - These consist of multiple linear equations with the same variables.

Example:

- 2x + 3y = 10

- 4x - 5y = 7

**2. Solving Linear Equations:**

a. **Isolating the Variable:** - The goal is to isolate the variable (e.g., **x**) on one side of the equation.

- Use inverse operations (addition, subtraction, multiplication, division) to do this.

b. **Balancing Equations:** - Perform the same operations on both sides of the equation to keep it balanced.

c. **Substitution:** - In some cases, you may need to substitute one expression for another to simplify the equation.

**3. Common Techniques for Solving Linear Equations:**

a. **Addition and Subtraction:** - Add or subtract constants to both sides of the equation to isolate the variable.

b. **Multiplication and Division:** - Multiply or divide both sides by constants to isolate the variable.

**4. Word Problems:**

- Linear equations are often used to model real-world scenarios. Translate the problem into an equation, solve it, and interpret the results in the context of the problem.

**5. Systems of Linear Equations:**

- In systems of linear equations, the goal is to find values of variables that satisfy all the equations simultaneously. Common methods include substitution, elimination, and matrices.

**6. Linear Equations in Slope-Intercept Form:**

- The equation
**y = mx + b**is in slope-intercept form, where:**y**and**x**are variables.**m**is the slope of the line.**b**is the y-intercept (the value of**y**when**x = 0**).

**7. Linear Equations in Point-Slope Form:**

- The equation
**y - y₁ = m(x - x₁)**represents a line in point-slope form, where:**y**and**x**are variables.**m**is the slope of the line.**(x₁, y₁)**is a point on the line.