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Algebra / Systems of Equations

Nonlinear Systems of Equations

A nonlinear system has at least one equation that isn't a straight line: a parabola, a circle, a hyperbola. That changes the answer to the first question you should ask, which is how many solutions to expect. Two lines meet at most once, but a line and a parabola can meet twice.

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Chapter 7 · Lesson 9 of 11Systems of Equations
  1. 1.Systems of Linear Equations
  2. 2.Substitution Method
  3. 3.Elimination Method
  4. 4.Graphing Systems of Equations
  5. 5.Systems of Equations With Three Variables
  6. 6.Augmented Matrices
  7. 7.Cramer's Rule
  8. 8.Inverse of a Matrix
  9. 9.Nonlinear Systems of Equations
  10. 10.Systems of Linear Inequalities
  11. 11.Systems of Equations Word Problems
See all algebra chapters →

Every system so far has been made of straight lines, and two straight lines can cross at most once. Swap one of them for a curve and that rule breaks. A line can slice through a parabola twice, just touch it once, or miss it entirely, and a circle and a hyperbola can cross four times. A nonlinear system of equations is any system with at least one equation like this. The methods are ones you already know, with substitution doing most of the work, but the first step is always the same: work out how many solutions to expect.

Substituting y = mx + k into y = ax² + bx + c gives the quadratic ax² + (b − m)x + (c − k) = 0
A line meeting a parabola always reduces to a quadratic

What Is a Nonlinear System?

A system is nonlinear when at least one of its equations is not of the form . Squared variables, products like , and square roots all make an equation nonlinear. Typical examples:

SystemShapes
and parabola and line
and circle and line
and circle and hyperbola
and hyperbola and line

As always, a solution is a point that satisfies every equation, which on a graph is a point where the curves intersect.

How Many Solutions Are There to This Nonlinear System?

Start with the picture. A line and a parabola can relate in three ways:

Three small graphs of a parabola and a line: the line cutting through the parabola at two points, the line tangent to the parabola at one point, and the line missing the parabola entirely
A line and a parabola: two points, one point of tangency, or none

Other pairs of shapes allow more crossings:

ShapesPossible numbers of solutions
line and parabola
line and circle
two different circles
two parabolas (or infinitely many if identical)
parabola and circle
circle and hyperbola

When a question asks “how many solutions are there to this nonlinear system?” and shows a graph, count the intersection points. If there’s no graph, substitute and count the real roots of the equation you get.

Counting Solutions With the Discriminant

For a line and a parabola, substituting always produces a quadratic . Its discriminant tells you the number of intersection points before you solve anything:

Number of solutionsPicture
positivethe line cuts through
zerothe line is tangent
negativethe line misses

Solving a Nonlinear System by Substitution

The steps are almost the same as for linear systems:

  1. Solve the simpler equation (usually the linear one) for one variable.
  2. Substitute into the nonlinear equation.
  3. Solve the resulting equation, usually a quadratic, by factoring or the quadratic formula.
  4. Find the partner value for each root using the equation from step 1.
  5. Check every point in both original equations.

Step 4 is the new part. A quadratic can give two roots, so there can be two solution points, and each needs its own partner.

Example 1: A Line and a Parabola

Solve and .

Step 1 — both equations give , so set them equal:

Step 2 — rearrange into standard form and factor:

Step 3 — two roots: or .

Step 4 — find each partner from :

Step 5 — check in the parabola: ✓ and ✓.

Answer: and

The parabola y = x² − 4 and the line y = x + 2 intersecting at (−2, 0) and (3, 5)
Two roots of the quadratic, two intersection points

Example 2: A Line and a Circle

Solve and .

Step 1 — substitute into the circle:

Step 2 — expand and simplify:

Step 3 — find each partner:

Step 4 — check in the circle: ✓ and ✓.

Answer: and

The circle x² + y² = 25 and the line y = x + 1 crossing at (3, 4) and (−4, −3)
The line enters the circle at one point and leaves at another

Remember to square the whole bracket: , not .

Example 3: One Solution (a Tangent Line)

Solve and .

Step 1 — set the expressions equal and rearrange:

Step 2 — a double root: , so .

Answer: one solution, . The discriminant is , and the line just touches the parabola.

No Solution

Solve and . Setting them equal gives , with discriminant . That’s negative, so there are no real roots and the line never meets the parabola. The system has no solution.

Solving a Nonlinear System by Elimination

When both equations have the same squared terms, elimination can be faster than substitution.

Example 4: A Circle and a Hyperbola

Solve and .

Step 1 — add the equations to eliminate :

Step 2 — substitute into the first equation:

Step 3 — combine: each works with each , because both equations only involve and .

Answer: four solutions, , , and

The circle x² + y² = 13 and the hyperbola x² − y² = 5 crossing at four points: (3, 2), (3, −2), (−3, 2) and (−3, −2)
A circle and a hyperbola can cross four times

Taking a square root gives two signs. Forgetting the is how most people find only one or two of these four points.

Worked Example: Area and Perimeter

A rectangle has an area of square meters and a perimeter of meters. Find its dimensions.

Step 1 — write the system, with length and width :

Step 2 — simplify the perimeter equation and solve for :

Step 3 — substitute into the area equation:

Step 4 — find the partners: gives , and gives .

Both solutions describe the same rectangle: 6 m by 8 m. Check: ✓ and ✓.

Common Mistakes to Avoid

  • Expecting one solution. Nonlinear systems routinely have two or more. Keep going after you find the first root.
  • Mismatching partners. Find each from its own . In Example 1, is not a solution.
  • Dropping the negative square root. means or .
  • Squaring a binomial wrongly. . The middle term is the one people forget.
  • Substituting into the wrong equation for the partner. Use the linear equation when you can. Substituting into a squared equation can introduce extra sign options that aren’t real solutions.

Practice Problems

Work each one before opening the answer.

Problem 1. How many solutions does the system and have? Find them.

Show answer

Step 1 — substitute :

Answer: two solutions, and

Problem 2. Solve and .

Show answer

Step 1 — set them equal and factor:

Step 2 — partners from : gives ; gives .

Answer: and

Problem 3. Solve and .

Show answer

Step 1 — substitute:

Step 2 — partners from : gives ; gives .

Answer: and

Problem 4. How many solutions does the system and have?

Show answer

Step 1 — set them equal:

Step 2 — both give .

Answer: two solutions, and

Problem 5. Solve and .

Show answer

Step 1 — solve the linear equation for and substitute:

Step 2 — partners: gives ; gives .

Answer: and

Problem 6. Does the line intersect the parabola ?

Show answer

Step 1 — set them equal:

Step 2 — the discriminant:

Answer: no, the system has no solution

Nonlinear systems complete the toolkit for solving equations together. The last step is using all of these methods where they matter most: turning a real situation into a system and solving it.

Frequently Asked Questions

What is a nonlinear system of equations?+

It is a system in which at least one equation is not linear, for example together with . The graphs include curves such as parabolas, circles or hyperbolas, and the solutions are the points where those graphs intersect.

How many solutions can a nonlinear system have?+

It depends on the shapes. A line and a parabola, or a line and a circle, meet in , or points. Two different circles also meet in at most . A circle and a parabola, or a circle and a hyperbola, can meet in up to points.

How do you solve a nonlinear system?+

Usually by substitution: solve the linear equation for one variable and substitute into the other equation, which produces a quadratic. Solve the quadratic, then find the matching value of the other variable for each root. When both equations contain and terms, elimination can cancel one of them instead.

How can you tell how many solutions a line and a parabola have without graphing?+

Substitute to get a quadratic and look at the discriminant . Positive means two intersection points, zero means one (the line is tangent), and negative means the line misses the parabola.

Why do you need to pair the x- and y-values carefully?+

Each solution is a point. When a quadratic gives two -values, each one has its own -value, found from the equation you substituted. Mixing an from one point with a from another gives a pair that is not on both graphs.

Related lessons

Substitution MethodThe Quadratic FormulaCircles: Centre, Radius, and the Equation of a CircleFactoring Quadratics