\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Solving Nonlinear Systems of Equations

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Types of Systems of Equations

Basics of linear systems with the same number of equations as unknowns - covered on the main linear systems page

Linear with fewer equations than unknowns - covered on the dependent systems page

Non-Linear - covered on this page

What Does It Mean to 'Solve'?

Like you did with linear systems, solving non-linear systems means to find the points that solve the both equations. In terms of graphs, these points are where the graphs intersect. We use the term non-linear when at least one of the equations is not a straight line.
Here is a great video where the instructor explains this in more detail.

Michel vanBiezen - What Does It Mean to "Solve..."?

video by Michel vanBiezen

Getting Started

Solving systems of nonlinear equations can be done using techniques you have already learned with linear equations, substitution and elimination. Knowing which one to use is based on the form of the equations and, if you carefully look at the systems, usually one of the techniques will seem to work best. Here are some ideas to get you started.

1. If you have terms that with powers, try elimination first. It is best not to solve for a variable under a power. For example, if one of your equations is \(y=x^2\), do not try to solve for x in order to use substitution. This will introduce a complexity that can lead to incomplete and incorrect answers.
2. If there is an obvious substitution, try to substitute cautiously. Sometimes substitution can get messy. Other times it can simplify the equations significantly.
3. You may end up with a quadratic. In this case, completing the square will help.

Difference From Linear Systems

Although we use the same techniques (substitution and elimination), we may end up with more than one solution. If you think about what is going on, this makes sense. For example, if we have a parabola and a line, intersection of the two curves may occur at two points. It is difficult to know just from the equations, how many points solve the system.

If you are allowed to, it helps to plot a graph on your calculator or computer and get a feel for what the graphs look like. This will help you know what to do when solving the equations.

Okay, now try your hand at some practice problems.

Schaum's Outline of Precalculus, 3rd Edition: 738 Solved Problems + 30 Videos

Practice

Unless otherwise instructed, find all the solutions to these nonlinear systems of equations, giving your answers in exact form.

These practice problems are divided into these sections.
1. Basic
2. Intermediate
3. Word Problems
4. Inequalities

Basic

\( x^2 + y^2 = 13 \) and \( 2x + y = -1 \)

Problem Statement

\( x^2 + y^2 = 13 \) and \( 2x + y = -1 \)

Final Answer

\( ( 1.2, -3.4 ) \) and \( ( -2, 3 ) \)

Problem Statement

\( x^2 + y^2 = 13 \) and \( 2x + y = -1 \)

Solution

Michel vanBiezen - 4116 video solution

video by Michel vanBiezen

Final Answer

\( ( 1.2, -3.4 ) \) and \( ( -2, 3 ) \)

Log in to rate this practice problem and to see it's current rating.

\( x^2+4x-y=7\) and \( 2x-y=-1 \)

Problem Statement

\( x^2+4x-y=7\) and \( 2x-y=-1 \)

Final Answer

\( (-4, -7) \) and \( (2, 5) \)

Problem Statement

\( x^2+4x-y=7\) and \( 2x-y=-1 \)

Solution

mattemath - 1781 video solution

video by mattemath

Final Answer

\( (-4, -7) \) and \( (2, 5) \)

Log in to rate this practice problem and to see it's current rating.

\(-x+y=4 \text{ and } x^2+y=3\)

Problem Statement

\(-x+y=4 \text{ and } x^2+y=3\)

Solution

mattemath - 1782 video solution

video by mattemath

Log in to rate this practice problem and to see it's current rating.

\(-2x+y=5 \text{ and } x^2+3x-y=1\)

Problem Statement

\(-2x+y=5 \text{ and } x^2+3x-y=1\)

Solution

mattemath - 1783 video solution

video by mattemath

Log in to rate this practice problem and to see it's current rating.

\( x^2 - y^2 = 1 \) and \( x^2 + y^2 = 49 \)

Problem Statement

\( x^2 - y^2 = 1 \) and \( x^2 + y^2 = 49 \)

Final Answer

\( ( 5, 2\sqrt{6} ) \), \( ( 5, -2\sqrt{6} ) \), \( ( -5, 2\sqrt{6} ) \) and \( ( -5, -2\sqrt{6} ) \)

Problem Statement

\( x^2 - y^2 = 1 \) and \( x^2 + y^2 = 49 \)

Solution

Michel vanBiezen - 4117 video solution

video by Michel vanBiezen

Final Answer

\( ( 5, 2\sqrt{6} ) \), \( ( 5, -2\sqrt{6} ) \), \( ( -5, 2\sqrt{6} ) \) and \( ( -5, -2\sqrt{6} ) \)

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{ \frac{x^2}{4} + \frac{y^2}{9} = 1 }\) and \( x^2 + y^2 = 9 \)

Problem Statement

\(\displaystyle{ \frac{x^2}{4} + \frac{y^2}{9} = 1 }\) and \( x^2 + y^2 = 9 \)

Final Answer

\( ( 0, 3 ) \) and \( ( 0, -3 ) \)

Problem Statement

\(\displaystyle{ \frac{x^2}{4} + \frac{y^2}{9} = 1 }\) and \( x^2 + y^2 = 9 \)

Solution

Michel vanBiezen - 4118 video solution

video by Michel vanBiezen

Final Answer

\( ( 0, 3 ) \) and \( ( 0, -3 ) \)

Log in to rate this practice problem and to see it's current rating.

\( 2x^2 - y^2 = 1 \) and \( xy = -1 \)

Problem Statement

\( 2x^2 - y^2 = 1 \) and \( xy = -1 \)

Final Answer

\( (1,-1) \) and \( ( -1,1 ) \)

Problem Statement

\( 2x^2 - y^2 = 1 \) and \( xy = -1 \)

Solution

Michel vanBiezen - 4119 video solution

video by Michel vanBiezen

Final Answer

\( (1,-1) \) and \( ( -1,1 ) \)

Log in to rate this practice problem and to see it's current rating.

\( x^2 + 2xy - 2y^2 = 6 \) and \( -x^2 + 3xy + 2y^2 = -6 \)

Problem Statement

\( x^2 + 2xy - 2y^2 = 6 \) and \( -x^2 + 3xy + 2y^2 = -6 \)

Final Answer

\( ( \sqrt{6}, 0 ) \) and \( ( -\sqrt{6}, 0 ) \)

Problem Statement

\( x^2 + 2xy - 2y^2 = 6 \) and \( -x^2 + 3xy + 2y^2 = -6 \)

Solution

Michel vanBiezen - 4120 video solution

video by Michel vanBiezen

Final Answer

\( ( \sqrt{6}, 0 ) \) and \( ( -\sqrt{6}, 0 ) \)

Log in to rate this practice problem and to see it's current rating.

\(y=x^2-9 \) and \( y=9-x^2\)

Problem Statement

\(y=x^2-9 \) and \( y=9-x^2\)

Solution

MIP4U - 1777 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

\(x^2-5y=6 \text{ and } x^2+y=18\)

Problem Statement

\(x^2-5y=6 \text{ and } x^2+y=18\)

Solution

MIP4U - 1778 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

\(2x^2-y^2=23 \text{ and } x^2+2y^2=34\)

Problem Statement

\(2x^2-y^2=23 \text{ and } x^2+2y^2=34\)

Solution

MIP4U - 1779 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

\(2x^2-10y^2=8 \text{ and } x^2-3y^2=6\)

Problem Statement

\(2x^2-10y^2=8 \text{ and } x^2-3y^2=6\)

Solution

Brightstorm - 1780 video solution

video by Brightstorm

Log in to rate this practice problem and to see it's current rating.

\( y=x^2 \text{ and } 3x+y=10 \)

Problem Statement

\( y=x^2 \text{ and } 3x+y=10 \)

Solution

Your Math Gal - 1789 video solution

video by Your Math Gal

Log in to rate this practice problem and to see it's current rating.

\( x^2+y^2=25 \text{ and } 3x+4y=0 \)

Problem Statement

\( x^2+y^2=25 \text{ and } 3x+4y=0 \)

Solution

Your Math Gal - 1790 video solution

video by Your Math Gal

Log in to rate this practice problem and to see it's current rating.

\( y-x=4 \text{ and } x^2+8x=y-16 \)

Problem Statement

\( y-x=4 \text{ and } x^2+8x=y-16 \)

Solution

Your Math Gal - 1791 video solution

video by Your Math Gal

Log in to rate this practice problem and to see it's current rating.

\( 6x-y=5 \text{ and } xy=1 \)

Problem Statement

\( 6x-y=5 \text{ and } xy=1 \)

Solution

Your Math Gal - 1792 video solution

video by Your Math Gal

Log in to rate this practice problem and to see it's current rating.

Find the solution(s) in the first quadrant to \( y = 2x^2 + 3 \) and \( y = -11x + 9 \)

Problem Statement

Find the solution(s) in the first quadrant to \( y = 2x^2 + 3 \) and \( y = -11x + 9 \)

Final Answer

\( ( 0.5, 3.5 ) \)

Problem Statement

Find the solution(s) in the first quadrant to \( y = 2x^2 + 3 \) and \( y = -11x + 9 \)

Solution

Michel vanBiezen - 4123 video solution

video by Michel vanBiezen

Final Answer

\( ( 0.5, 3.5 ) \)

Log in to rate this practice problem and to see it's current rating.

\( x^2 + y^2 = 25 \) and \( y = 4x/3 \)

Problem Statement

\( x^2 + y^2 = 25 \) and \( y = 4x/3 \)

Final Answer

\( ( 3,4 ) \) and \( ( -3,-4 ) \)

Problem Statement

\( x^2 + y^2 = 25 \) and \( y = 4x/3 \)

Solution

yaymath - 4124 video solution

video by yaymath

Final Answer

\( ( 3,4 ) \) and \( ( -3,-4 ) \)

Log in to rate this practice problem and to see it's current rating.

\( y = -x^2 + 4 \) and \( y = 2x + 1 \)

Problem Statement

\( y = -x^2 + 4 \) and \( y = 2x + 1 \)

Final Answer

\( ( -3,-5 ) \) and \( ( 1,3 ) \)

Problem Statement

\( y = -x^2 + 4 \) and \( y = 2x + 1 \)

Solution

yaymath - 4125 video solution

video by yaymath

Final Answer

\( ( -3,-5 ) \) and \( ( 1,3 ) \)

Log in to rate this practice problem and to see it's current rating.

\( x^2 + y^2 = 45 \) and \( y^2 - x^2 = 27 \)

Problem Statement

\( x^2 + y^2 = 45 \) and \( y^2 - x^2 = 27 \)

Final Answer

4 solutions, all combinations of \( ( \pm 3, \pm 6 ) \)

Problem Statement

\( x^2 + y^2 = 45 \) and \( y^2 - x^2 = 27 \)

Solution

yaymath - 4126 video solution

video by yaymath

Final Answer

4 solutions, all combinations of \( ( \pm 3, \pm 6 ) \)

Log in to rate this practice problem and to see it's current rating.

\( y=x/2 \text{ and } 2x^2-y^2=7 \)

Problem Statement

\( y=x/2 \text{ and } 2x^2-y^2=7 \)

Solution

Khan Academy - 1784 video solution

video by Khan Academy

Log in to rate this practice problem and to see it's current rating.

\( y=x+1 \) and \( x^2+y^2=25 \)

Problem Statement

\( y=x+1 \) and \( x^2+y^2=25 \)

Solution

Khan Academy - 1785 video solution

video by Khan Academy

Log in to rate this practice problem and to see it's current rating.

\( y=-x^2+6 \text{ and }y=-2x-2 \)

Problem Statement

\( y=-x^2+6 \text{ and }y=-2x-2 \)

Solution

Khan Academy - 1786 video solution

video by Khan Academy

Log in to rate this practice problem and to see it's current rating.

\( y=2x^2+3x-6 \text{ and } y=-x^2 \)

Problem Statement

\( y=2x^2+3x-6 \text{ and } y=-x^2 \)

Solution

Khan Academy - 1787 video solution

video by Khan Academy

Log in to rate this practice problem and to see it's current rating.

\( y=2(x-4)^2+3 \text{ and } y=-x^2+2x-2 \)

Problem Statement

\( y=2(x-4)^2+3 \text{ and } y=-x^2+2x-2 \)

Solution

Khan Academy - 1788 video solution

video by Khan Academy

Log in to rate this practice problem and to see it's current rating.

Intermediate

\( x^3 + 9x^2y = 10 \) and \( y^3 + xy^2 = 2 \)

Problem Statement

\( x^3 + 9x^2y = 10 \) and \( y^3 + xy^2 = 2 \)

Hint

Expand out \( (x + 3y)^3 \).

Problem Statement

\( x^3 + 9x^2y = 10 \) and \( y^3 + xy^2 = 2 \)

Hint

Expand out \( (x + 3y)^3 \).

Solution

blackpenredpen - 3784 video solution

video by blackpenredpen

Log in to rate this practice problem and to see it's current rating.

Word Problems

The area of a rectangle is \( 8 \) and the perimeter is 12. What are the dimensions?

Problem Statement

The area of a rectangle is \( 8 \) and the perimeter is 12. What are the dimensions?

Final Answer

\( ( 2,4 ) \) and \( ( 4,2 ) \)

Problem Statement

The area of a rectangle is \( 8 \) and the perimeter is 12. What are the dimensions?

Solution

Michel vanBiezen - 4121 video solution

video by Michel vanBiezen

Final Answer

\( ( 2,4 ) \) and \( ( 4,2 ) \)

Log in to rate this practice problem and to see it's current rating.

The sum of the first number squared and the second number squared equals 73. The difference between second number squared and three times the first number squared equals 37. Find all possible pairs of numbers.

Problem Statement

The sum of the first number squared and the second number squared equals 73. The difference between second number squared and three times the first number squared equals 37. Find all possible pairs of numbers.

Final Answer

\( ( \pm 3, \pm 8 ) \)

Problem Statement

The sum of the first number squared and the second number squared equals 73. The difference between second number squared and three times the first number squared equals 37. Find all possible pairs of numbers.

Solution

Michel vanBiezen - 4122 video solution

video by Michel vanBiezen

Final Answer

\( ( \pm 3, \pm 8 ) \)

Log in to rate this practice problem and to see it's current rating.

Inequalities

\( x^2 + y^2 \leq 16 \) and \( x^2 - y^2 \gt 9 \)
For this problem, do not evaluate points. Just graph and shade the area(s) that represent your answer.

Problem Statement

\( x^2 + y^2 \leq 16 \) and \( x^2 - y^2 \gt 9 \)
For this problem, do not evaluate points. Just graph and shade the area(s) that represent your answer.

Solution

yaymath - 4127 video solution

video by yaymath

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Precalculus

Log in to rate this page and to see it's current rating.

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

memorize to learn

Shop Amazon - Rent eTextbooks - Save up to 80%

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Getting Started

Difference From Linear Systems

Practice

Practice Search

Practice Instructions

Unless otherwise instructed, find all the solutions to these nonlinear systems of equations, giving your answers in exact form.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics