Linear Equations

Linear equations are a fundamental concept in algebra and form the basis for understanding more complex mathematical ideas.

Definition

A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable. It can be written in the form y = mx + b, where:

'y' is the dependent variable,
'x' is the independent variable,
'm' is the slope or gradient of the line,
'b' is the y-intercept, which is the value of y when x = 0.

For example, in the equation y = 2x + 1, 2 is the slope and 1 is the y-intercept.

Properties

A linear equation forms a straight line when graphed. The slope m determines the steepness of the line and the y-intercept b determines where the line crosses the y-axis. The x-intercept is the value of x when y is equal to zero.

Applications

Linear equations are used extensively in both pure and applied mathematics, as well as in fields like physics, engineering, economics, computer science, and data analysis. They are used to model relationships between variables, solve problems, and make predictions.