A linear equation is an equation of the form:

$ ax + b = 0 $, where $ a \neq 0 $.

In this equation:

• $ x $ is the variable.
• $ a $ is the coefficient of $ x $.
• $ b $ is a constant term.

Linear equations can be solved using various methods. The solution to a linear equation is the value of $ x $ that makes the equation true.

Types of Linear Equations

Linear equations can be categorized into different types based on the number of variables:

• Single Variable: An equation in one variable, e.g., $ 2x + 3 = 7 $.
• Two Variables: An equation in two variables, e.g., $ 3x + 4y = 12 $.

Graphical Representation

A linear equation in two variables can be represented graphically as a straight line on a Cartesian plane. The general form is:

$ Ax + By + C = 0 $.

The slope-intercept form is:

$ y = mx + c $, where:

• $ m $ is the slope of the line.
• $ c $ is the y-intercept.

Methods to Solve Linear Equations

There are several methods to solve linear equations:

• Graphical Method: Plotting the equations on a graph to find the point of intersection.
• Substitution Method: Solving one equation for a variable and substituting into the other equation.
• Elimination Method: Adding or subtracting equations to eliminate one variable.

Example of a Single Variable Linear Equation

Consider the equation:

$ 2x - 4 = 0 $.

To solve for $ x $:

1. Add 4 to both sides: $ 2x = 4 $.
2. Divide by 2: $ x = 2 $.

Example of a Two Variable Linear Equation

Consider the system of equations:

• $ 2x + 3y = 6 $
• $ x - y = 1 $

Using the substitution method:

1. From the second equation, solve for $ x $: $ x = y + 1 $.
2. Substitute into the first equation: $ 2(y + 1) + 3y = 6 $.
3. Simplify: $ 2y + 2 + 3y = 6 $.
4. Combine like terms: $ 5y + 2 = 6 $.
5. Subtract 2 from both sides: $ 5y = 4 $.
6. Divide by 5: $ y = \frac{4}{5} $.

Now substitute back to find $ x $:

1. $ x = \frac{4}{5} + 1 = \frac{9}{5} $.

Applications of Linear Equations

Linear equations are widely used in various fields such as:

• Business for profit and loss calculations.
• Science for calculating relationships between variables.
• Engineering for designing structures.