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

17Calculus Precalculus - Solving Nonlinear Systems of Equations

Algebra

Polynomials

Functions

Rational Functions

Graphing

Matrices

Systems

Trigonometry

Complex Numbers

Applications

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Articles

Algebra

Functions

Functions

Polynomials

Rational Functions

Graphing

Matrices & Systems

Matrices

Systems

Trigonometry & Complex Numbers

Trigonometry

Complex Numbers

Applications

SV Calculus

MV Calculus

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Articles

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

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.

Practice

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

Basic

\(y=x^2-9 \text{ and } y=9-x^2\)

Problem Statement

\(y=x^2-9 \text{ and } y=9-x^2\)

Solution

1777 video

video by MIP4U

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

Problem Statement

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

Solution

1778 video

video by MIP4U

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

Problem Statement

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

Solution

1779 video

video by MIP4U

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

Problem Statement

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

Solution

1780 video

video by Brightstorm

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

Problem Statement

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

Solution

1781 video

video by mattemath

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

Problem Statement

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

Solution

1782 video

video by mattemath

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

Problem Statement

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

Solution

1783 video

video by mattemath

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

Problem Statement

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

Solution

1789 video

video by Your Math Gal

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

Problem Statement

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

Solution

1790 video

video by Your Math Gal

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

Problem Statement

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

Solution

1791 video

video by Your Math Gal

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

Problem Statement

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

Solution

1792 video

video by Your Math Gal

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

Problem Statement

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

Solution

1784 video

video by Khan Academy

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

Problem Statement

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

Solution

1785 video

video by Khan Academy

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

Problem Statement

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

Solution

1786 video

video by Khan Academy

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

Problem Statement

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

Solution

1787 video

video by Khan Academy

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

Problem Statement

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

Solution

1788 video

video by Khan Academy

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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

3784 video

video by blackpenredpen

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Really UNDERSTAND Precalculus

Topics You Need To Understand For This Page

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

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Getting Started

Difference From Linear Systems

Practice

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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.

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