Linear Equations

Linear equations form the foundation of many algebraic and real-world applications, characterized by their simplicity and utility. A linear equation is an algebraic statement in which each term is either a constant or a product of a constant and a single variable. These equations represent straight lines when plotted on a Cartesian coordinate plane, reflecting a constant rate of change between two quantities.

A typical linear equation in one variable has the form: ax + b = c, where a, b, and c are constants and x is the variable. The solution process isolates the variable to determine its value, making the equation true. This is done through operations such as addition, subtraction, multiplication, or division, adhering to the principles of maintaining equality on both sides.

Linear equations are frequently used to model financial scenarios, such as determining cost structures. Consider a cab company that imposes a flat fee plus a constant rate per kilometer. If a 10-kilometer ride costs $35, and the flat fee is known, the unknown per-kilometer rate can be modeled as a linear equation. Let the rate per kilometer be r, and the flat fee is $5. The equation becomes:

Subtracting the flat fee and solving for r:

Thus, the rate per kilometer is $3. Substituting back confirms the solution: 10 × 3 + 5 = 35. This example illustrates how linear equations are essential tools for quantifying relationships and solving practical problems efficiently.