Nonlinear Equation — Definition, Formula & Examples

A nonlinear equation is any equation that does not form a straight line when graphed. It contains at least one variable raised to a power other than 1, or variables multiplied together, or variables inside functions like square roots or absolute values.

An equation in one or more variables is nonlinear if it cannot be written in the form $a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$ , where $a_i$ and $b$ are constants. Equivalently, at least one variable appears with an exponent other than 1, inside a nonlinear function, or in a product with another variable.

How It Works

To determine whether an equation is nonlinear, examine every term that contains a variable. If any variable is squared, cubed, under a radical, in a denominator, inside a trigonometric function, or multiplied by another variable, the equation is nonlinear. For example,

y = 3x + 5

is linear because

x

appears only to the first power with a constant coefficient. But

y = x^2 + 3x + 5

is nonlinear because of the

x^2

term. The key test: can you rewrite the equation so every variable term is just a constant times a single variable to the first power?

Worked Example

Problem: Classify each equation as linear or nonlinear: (a)

y = 4x - 7

, (b)

y = x^2 + 1

, (c)

xy = 12

Equation (a): In

y = 4x - 7

, both

y

and

x

appear to the first power with no products of variables. This is linear.

y = 4x - 7 \quad \text{(linear)}

Equation (b): In

y = x^2 + 1

, the variable

x

is raised to the second power. This makes the equation nonlinear.

y = x^2 + 1 \quad \text{(nonlinear — contains } x^2\text{)}

Equation (c): In

xy = 12

, two variables are multiplied together. A product of variables is not a first-degree term, so this is nonlinear.

xy = 12 \quad \text{(nonlinear — product of variables)}

Answer: (a) Linear, (b) Nonlinear, (c) Nonlinear.

Why It Matters

Recognizing nonlinear equations matters because they require different solving techniques than linear ones. In physics, projectile motion follows

y = -\frac{1}{2}gt^2 + v_0t + h_0

, a nonlinear equation whose parabolic shape you need to analyze using quadratic methods rather than simple slope-intercept reasoning.

Common Mistakes

Mistake: Thinking

y = \frac{x}{3} + 2

is nonlinear because of the fraction.

Correction: Dividing a variable by a constant is the same as multiplying by a constant:

\frac{x}{3} = \frac{1}{3}x

. The variable is still to the first power, so the equation is linear. An equation is only nonlinear when a variable appears in the denominator (like

y = \frac{1}{x}

), at a non-first-power exponent, or in a product with another variable.