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

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

video by MIP4U 

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Problem Statement 

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

video by MIP4U 

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Problem Statement 

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

video by MIP4U 

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Problem Statement 

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

video by Brightstorm 

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Problem Statement 

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

video by mattemath 

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Problem Statement 

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

video by mattemath 

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Problem Statement 

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

video by mattemath 

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Problem Statement 

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

video by Your Math Gal 

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Problem Statement 

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

video by Your Math Gal 

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Problem Statement 

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

video by Your Math Gal 

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Problem Statement 

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

video by Your Math Gal 

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Problem Statement 

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

video by Khan Academy 

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Problem Statement 

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

video by Khan Academy 

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Problem Statement 

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

video by Khan Academy 

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Problem Statement 

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

video by Khan Academy 

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Problem Statement 

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

video by Khan Academy 

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