Linear Equations Fundamentals

Study Notes

Linear Equations

Linear equations are mathematical expressions consisting of constants and variables raised only to the first power. They can represent relationships between quantities in various situations, such as physics, engineering, economics, biology, chemistry, demographics, and business. In linear algebra, a set of these equations is called linear system.

Formal Definition

A linear equation is typically written in one of two ways: graphically or symbolically. Graphically, it's represented by plotting lines on coordinate axes. Symbolically, a linear equation is expressed using one variable with all other terms constant. For example, ax + b = c, where a represents the coefficient, x represents the dependent variable, b represents the y-intercept, and c represents the equilibrium point or point of intersection.

Slope-Intercept Form

In slope-intercept form, a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept. The slope determines how much the output changes when the input increases by one unit. If the slope is positive, the relationship is increasing; if negative, decreasing; if zero, there's no change.

Applications

Linear equations have widespread applications across numerous disciplines. For instance, they are used in physics for modeling projectile motion using kinematic equations. In the field of economics, they help determine price elasticity of demand, which measures responsiveness of quantity demanded to price changes. Similarly, in statistics, linear regression models allow researchers to make predictions based on observed data points.

Moreover, linear equations play a crucial role in computer graphics, where they define objects like roads, buildings, and landscapes. They also find application in network graphs, telecommunications networks, electrical circuits, and queuing systems.

Solving Linear Equations

To solve a linear equation, you need to isolate the variable on one side of the equal sign. This often involves performing inverse operations like addition, subtraction, multiplication, and division until you get the desired result. Sometimes, solving a linear equation might involve eliminating one of the variables through the process of substitution or elimination method.

For example, consider the following equation: 2x - 3 = 7. To isolate x, we add 3 to both sides of the equation: 2x - 3 + 3 = 7 + 3 → 2x = 10. Then, divide both sides by 2: (2x)/2 = 10/2 → x = 5.

The solution to this equation is x = 5.

Description

Learn the basics of linear equations, including their formal definitions, representations, applications in various fields such as physics and economics, and techniques for solving them. Understand key concepts like slope-intercept form and how to isolate variables to find solutions.