System of Nonlinear Equations — Definition, Formula & Examples

A system of nonlinear equations is a set of two or more equations, solved together, where at least one equation is not linear. Its graph can include curves such as circles or parabolas, and every point where all the graphs meet is a solution.

A system of equations in variables $x_1, \dots, x_n$ is nonlinear if at least one equation cannot be written in the form $a_1x_1 + a_2x_2 + \dots + a_nx_n = b$ with constants $a_i$ and $b$ . A solution is an ordered $n$ -tuple that satisfies every equation in the system at the same time. Over the real numbers, the solution set is the intersection of the graphs of the equations.

How It Works

Substitution is usually the best method. Solve the simpler equation (often the linear one) for one variable, then substitute that expression into the other equation. This produces a single equation in one variable, often a quadratic. Solve it, then back-substitute each value to find the matching value of the other variable. Graphically, each real solution is an intersection point. For example, a line and a circle meet at 0, 1, or 2 points.

Worked Example

Problem: Solve the system

x^2 + y^2 = 25

and

y = x + 1

Step 1: The second equation already gives y in terms of x. Substitute

x + 1

for y in the circle equation, then expand.

x^2 + (x+1)^2 = 25 \;\Rightarrow\; 2x^2 + 2x + 1 = 25

Step 2: Subtract 25 from both sides, then divide every term by 2. Factor the quadratic and solve for x.

2x^2 + 2x - 24 = 0 \;\Rightarrow\; x^2 + x - 12 = 0 \;\Rightarrow\; (x+4)(x-3) = 0 \;\Rightarrow\; x = -4 \text{ or } x = 3

Step 3: Back-substitute each x-value into the linear equation

y = x + 1

to get its y-value.

x = 3: \; y = 4 \qquad x = -4: \; y = -3

Answer: The solutions are

(3, 4)

and

(-4, -3)

. On a graph, these are the two points where the line crosses the circle of radius 5.

Why It Matters

In Algebra 2 and precalculus, you use these systems to find where a line crosses a parabola or circle. Physics and engineering problems use them too, such as finding when a thrown ball reaches a given height or where two orbits cross.

Common Mistakes

Mistake: Finding y by substituting x back into the nonlinear equation and keeping every result. With

x = 3

x^2 + y^2 = 25

, you get

y = \pm 4

, so the false pair

(3, -4)

sneaks in.

Correction: Back-substitute into the linear equation, or check every pair in both original equations. The pair

(3, -4)

fails

y = x + 1

because

3 + 1 = 4

, not

-4

Related Terms

System of Equations — the broader concept this term builds on
Linear Meaning in Math — Defines what makes an equation linear or not
Back-Substitution — Final step for finding the second variable
Consistent System of Equations — System with at least one solution
Inconsistent System of Equations — System whose graphs never intersect
Equivalent Systems of Equations — Systems that share the same solution set
Equation of a Line: All 4 Forms with Formulas & Examples — The linear equation often used for substitution