The Difference Between Linear & Nonlinear Equations

This illustration shows an example of a nonlinear graph.
••• graph image by Roman Sigaev from Fotolia.com

In the world of mathematics, there are several types of equations that scientists, economists, statisticians and other professionals use to predict, analyze and explain the universe around them. These equations relate variables in such a way that one can influence, or forecast, the output of another. In basic mathematics, linear equations are the most popular choice of analysis, but nonlinear equations dominate the realm of higher math and science.

Types of Equations

Each equation gets its form based on the highest degree, or exponent, of the variable. For instance, in the case where y = x³ – 6x + 2, the degree of 3 gives this equation the name “cubic.” Any equation that has a degree no higher than 1 receives the name “linear.” Otherwise, we call an equation “nonlinear,” whether it is quadratic, a sine-curve or in any other form.

Input-Output Relationships

In general, “x” is considered to be the input of an equation and “y” is considered to be the output. In the case of a linear equation, any increase in “x” will either cause an increase in “y” or a decrease in “y” corresponding to the value of the slope. In contrast, in a nonlinear equation, “x” may not always cause “y” to increase. For example, if y = (5 – x)², “y” decreases in value as “x” approaches 5, but increases otherwise.

Graph Differences

A graph displays the set of solutions for a given equation. In the case of linear equations, the graph will always be a line. In contrast, a nonlinear equation may look like a parabola if it is of degree 2, a curvy x-shape if it is of degree 3, or any curvy variation thereof. While linear equations are always straight, nonlinear equations often feature curves.

Exceptions

Except for the case of vertical lines (x = a constant) and horizontal lines (y = a constant), linear equations will exist for all values of “x” and “y.” Nonlinear equations, on the other hand, may not have solutions for certain values of “x” or “y.” For instance, if y = sqrt(x), then “x” exists only from 0 and beyond, as does “y,” because the square root of a negative number does not exist in the real number system and there are no square roots that result in a negative output.

Benefits

Linear relationships can be best explained by linear equations, where the increase in one variable directly causes the increase or decrease of another. For example, the number of cookies you eat in a day could have a direct impact on your weight as illustrated by a linear equation. However, if you were analyzing the division of cells under mitosis, a nonlinear, exponential equation would fit the data better.

For more tips on distinguishing between the two, watch the video below:

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