What Are Linear Equations & When Do They Apply?

By Society & Culture & Entertainment Last updated Tuesday, November/05/2024

Definition

A linear equation is any equation of the form Ax + By = C, where A, B and C are numbers and x and y are variables. Ax + By + C is known as the standard form of a linear equation, but the most common form is slope-intercept form: y = mx + b, where x is the independent variable and y is the dependent variable. M is the slope (how much y increases or decreases when x increases by 1) and b is the y-intercept (the value of y when x is equal to zero).

You can gather a lot of information about the behavior of a linear equation just by examining the values. For example, if the equation y = 5x + 2 relates the change in the price of gas in dollars every year since 2000, you know that the change in price has been constant; in other words, every year the price has gone up by the same amount. You also know the slope of the equation is 5 because m is equal to 5, which means that every year the price change is an increase of 5 dollars. Finally, you know that the y-intercept is 2, meaning the y value (cost of gas) when x is equal to zero (the year 2000) is $2.

To graph linear equations, identify the y-intercept and slope from the equation y = mx + b. Mark a dot on the y-axis at the point corresponding to the value of b. Then mark a dot exactly one unit to the right (where x = 1) and m units up or down (up if m is positive, down if m is negative). For example, if b = 3 and m = 2, you would mark a dot at (0, 3) and then move one unit to the right and two units up and mark a dot at (1, 5). Draw a line connecting these two dots and extending to infinity in either direction. This graph tells you the value of y for every value of x.

The application of linear equations includes any case where two variables have a proportionate, or linear, relationship. For example, if a basketball player moves away from a hoop, she is proportionately less likely to make the shot at that distance. You could use a linear equation to describe this relationship, where x is the distance to the basket and y is the percentage of shots made. In the equation y = mx + b, b would be the percentage of shots made at point-blank range and m would be the decrease in shooting percentage with every foot she moves away from the basket.