Linear Equations

A linear equation is a first-degree algebraic equation, meaning an equation where the variable only appears to the first power (i.e., not raised to higher powers) and does not appear in products with other variables.

The general form of a linear equation with one variable $ x $ is:

$$ ax + b = 0 $$

Where $ a $ and $ b $ are coefficients (with $ a \neq 0 $ to ensure the equation is truly linear) while $ x $ is the variable.

A linear equation has the following characteristics:

First Degree
The variable $ x $ is first degree, meaning it appears only as $ x $ and not as $ x^2 $, $ x^3 $, etc.
Non-zero Coefficient
The coefficient $ a $ that multiplies the variable $ x $ is not zero ($ a \neq 0 $).
Unique Solution
If the equation is in the standard form $ ax + b = 0 $, it has a unique solution given by $ x = -\frac{b}{a} $.

Examples of Linear Equations

Here are some examples of linear equations with one unknown variable:

$$ 2x + 3 = 0 $$

Solution: $ x = -\frac{3}{2} $

$$ 5x + 10 = 0 $$

Solution: $ x = -2 $

$$ x - 7 = 0 $$

Solution: $ x = 7 $

Linear Equations with Multiple Variables

A linear equation can also involve more than one variable.

For example, with two variables $ x $ and $ y $, the general form is:

$$ ax + by + c = 0 $$

Where $ a $, $ b $, and $ c $ are coefficients.

Non-linear Equations. An equation is non-linear if the variable appears raised to a power greater than one or if it appears in products or non-linear functions. For example, the following equation is not linear: $ x^2 + 2x + 1 = 0 $ because the variable is raised to the second power (quadratic equation).

In conclusion, a linear equation is a first-degree algebraic equation that can be represented in the form $ ax + b = 0 $ for a single variable, can involve multiple variables, and is characterized by having a single solution determined by the equation itself.