CITE THIS NOTEBOOK: Solving a system of nonlinear equations by Athanasios Paraskevopoulos. Wolfram Community FEB 18 2023.
In this notebook, we will consider the intersection of a hyperbola and a line, a circle and a line, and a circle and a hyperbola . The methods for solving systems of nonlinear equations are similar to those for linear equations .
A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear . Recall that a linear equation can take the form Ax+By=C. Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.
For example, suppose we ask for two numbers with a sum of 13 and a sum of squares of 89.​If x, y are the two numbers , then x + y = 13 and
2
x
+
2
y
= 89. Since we are also looking for common solutions of the two equations. To solve the system we work as follows :

Intersection of a Circle and a Line

In[]:=
syst={x+y==13,x^2+y^2==89}
Out[]=
{x+y13,
2
x
+
2
y
89}
In[]:=
sol1=Solve[sys,{x,y}]
Out[]=
{{x5,y8},{x8,y5}}
The answer to the problem, of course, is that the required numbers are 5 and 8.
In[]:=
Plot-x+13,
89-
2
x
,-
89-
2
x
,{x,-10,20},PlotRange->All,AspectRatio->Automatic,AxesLabel->{x,y},PlotStyle->{Blue,Black,Black},PlotLegends->"Expressions"
Out[]=
-x+13
89-
2
x

Intersection of a Hyperbola and a Line

Solve the system x + y = 5
x * y = 6
In[]:=
sys={x+y==5,x*y==6}
Out[]=
{x+y5,xy6}
In[]:=
sol=Solve[sys,{x,y}]
Out[]=
{{x2,y3},{x3,y2}}
So the system has two solutions (2, 3) and (3, 2) .
The first equation of the system x + y = 5 represents a line, while the second equation x*y = 6 represents the hyperbola y
=
6
x
. Therefore the coordinates of the common points of the line and the hyperbola will give us the solutions of the system .
In[]:=
Plot[{-x+5,6/x},{x,-7,7},PlotRange->Automatic,PlotLegends->"Expressions"]
Out[]=
-6
-4
-2
2
4
6
-10
10
20
-x+5
6
x

Intersection of a Hyperbola and a Circle

Solve the system
2
x
+
2
y
= 13 x * y = 6
In[]:=
system={x^2+y^2==13,x*y==6}
Out[]=
{
2
x
+
2
y
13,xy6}
In[]:=
soln=Solve[system,{x,y}]
Out[]=
{{x-3,y-2},{x-2,y-3},{x2,y3},{x3,y2}}
So the system has four solutions (3, 2), (-3, -2), (2, 3) and (-2, -3) .
In[]:=
Plot6x,
13-
2
x
,-
13-
2
x
,{x,-10,10},PlotRange->Automatic,AspectRatio->Automatic,PlotStyle->{Blue,Black,Black},PlotLegends->"Expressions"
Out[]=
-10
-5
5
10
-10
-5
5
10
6
x
13-
2
x