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 
NonLinear  covered on this page 
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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
\(y=x^29 \text{ and } y=9x^2\)
Problem Statement 

\(y=x^29 \text{ and } y=9x^2\)
Solution 

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\(x^25y=6 \text{ and } x^2+y=18\)
Problem Statement 

\(x^25y=6 \text{ and } x^2+y=18\)
Solution 

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

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

video by MIP4U 

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\(2x^210y^2=8 \text{ and } x^23y^2=6\)
Problem Statement 

\(2x^210y^2=8 \text{ and } x^23y^2=6\)
Solution 

video by Brightstorm 

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\(x^2+4xy=7 \text{ and } 2xy=1\)
Problem Statement 

\(x^2+4xy=7 \text{ and } 2xy=1\)
Solution 

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 

video by mattemath 

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

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

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 

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 

video by Your Math Gal 

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\( yx=4 \text{ and } x^2+8x=y16 \)
Problem Statement 

\( yx=4 \text{ and } x^2+8x=y16 \)
Solution 

video by Your Math Gal 

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\( 6xy=5 \text{ and } xy=1 \)
Problem Statement 

\( 6xy=5 \text{ and } xy=1 \)
Solution 

video by Your Math Gal 

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

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

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 

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\( y=x^2+6 \text{ and }y=2x2 \)
Problem Statement 

\( y=x^2+6 \text{ and }y=2x2 \)
Solution 

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\( y=2x^2+3x6 \text{ and } y=x^2 \)
Problem Statement 

\( y=2x^2+3x6 \text{ and } y=x^2 \)
Solution 

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\( y=2(x4)^2+3 \text{ and } y=x^2+2x2 \)
Problem Statement 

\( y=2(x4)^2+3 \text{ and } y=x^2+2x2 \)
Solution 

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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 

video by blackpenredpen 

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Really UNDERSTAND Precalculus
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  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle 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
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
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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Practice Instructions
Unless otherwise instructed, find all the solutions to these nonlinear systems of equations, giving your answers in exact form.