Remember the methods used to solve systems of linear equations. How would you adapt them to solve a nonlinear system?
See solution.
Practice makes perfect
There are many ways to solve a nonlinear system of equations. We will mention some of them and describe how they work briefly. We will talk about each method one at a time.
Solving the System by Graphing
As with any other system of equations, we can graph both functions together. The intersection points represent the solutions to the system.
Solving the System by Using an Analytical Approach
y = x^2+ 4x+10 & (I) - y = x^2+ 6x+10 & (II)
y = x^2+ 4x+10 [-0.55em]
+ 3.4cm [-0.55em]
- y = x^2+ 6x+10
0 = 2x^2+10x+20
Once we have solved the resulting quadratic equation, we can go back to the original system and find the remaining missing variable.
Solving the System by Using a Numerical Approach
We could also use tables to solve the system. We do this by making a table of values for x and for the system's functions. With it, we can identify for which x-values the functions have the same value.
y=x^2 & (I) y=- x^2+2 & (II)
x
y=x^2
y=- x^2+2
- 2
y = ( - 2)^2 = 4
y = - ( - 2)^2 + 2= 2
- 1
y = ( - 1)^2 = 1
y = - ( - 1)^2 + 2= 1
0
y = ( 0)^2 = 0
y = - ( 0)^2 + 2= 2
1
y = ( 1)^2 = 1
y = - ( 1)^2 + 2= 1
2
y = ( 2)^2 = 4
y = - ( 2)^2 + 2= 2
This method is not always exact, but we can improve the precision as needed by taking smaller intervals between the x-values used.
