A linear relationship can be represented by a straight line. The property of being linear can also be referred to as linearity. Linearity is closely related to proportionality. The relationship between weight and mass is an example of linearity. They are directly proportional based on gravity.
Identifying linear equations
In algebra, linear equations are equations that, as the name suggests, exhibit linearity. There are a number of ways to identify a linear equation. Generally, a linear equation contains constants and variables and can be written in the form y = mx + b.
There are also many different forms of a linear equation. Some of the most common forms include slope-intercept form, point-slope form, and standard form. Being familiar with some of these forms can make it easier to recognize when we are working with linear equations.
Linear equations are first order equations, which means that the variables in a linear equation must have a degree of 1. This means that any exponent on the variables in the equation cannot be larger than 1. Cube roots (x⅓), squares (x2), square roots (x½), absolute values (|x|), and other more complex functions are non-linear. A linear equation can be also be thought of as a first degree polynomial. We can check if an equation is linear by simplifying it as much as possible and confirming that the variables in the equation have a degree of 1.
1. y = 3x + 5
This is a linear equation with a slope of 3 and a y-intercept at (0, 5). Though it isn't written, the exponent on the x is 1.
2. 2x2 + 3y + 12 = 0
The equation above is not a linear equation because it is a 2nd degree polynomial since there is an exponent of 2 on x.
In many cases, once you are familiar with the various forms of a linear equation, it should be relatively easy to identify whether an equation is linear based on the equation. However, graphing the equation, whether doing it by hand or with a calculator, can also allow us to quickly visually identify whether an equation is linear. Plotting a few points may often be enough to see whether the graph curves or has any properties that make it non-linear.
Plugging in points
Given an equation, we can also plug in values for x. A function that has a linear relationship will have a constant slope. We can test this by plugging in values for x that have a constant difference between them (2, 4, and 6 have a difference of 2), then determining the y value. If the difference between the y values is the same, the equation has a constant slope and is linear. Otherwise, the equation is not linear.
Test whether y = 3x + 5 is linear.
Plugging in 2, 4, and 6, which have a difference of 2 between each number:
y = 3(2) + 5 = 11
y = 3(4) + 5 = 17
y = 3(6) + 5 = 23
There is a difference of 6 between 11 and 17 and 17 and 23. Since the slope is constant (5/2), the equation is linear.