# 1107 MrG How do we solve a Non Linear System?

## 2787 days ago by MATH4R2013

#1) find all POIs: 3*x-y==-2, 2*x^2-y==0 var('y') show(solve(3*x-y==-2,y)) show(solve(2*x^2-y==0,y))
 \newcommand{\Bold}{\mathbf{#1}}\left[y = 3 \, x + 2\right] \newcommand{\Bold}{\mathbf{#1}}\left[y = 2 \, x^{2}\right]
solve([3*x-y==-2, 2*x^2-y==0],x,y)
 $\newcommand{\Bold}{\mathbf{#1}}\left[\left[x = 2, y = 8\right], \left[x = \left(-\frac{1}{2}\right), y = \left(\frac{1}{2}\right)\right]\right]$
plot([3*x+2,2*x^2],-1,3)  plot([3*x+2,2*x^2],-1,3,aspect_ratio=1)  #2) Find all POIs: x^2+y^2==13, x^2-y==7 show(solve(x^2+y^2==13,y)) show(solve(x^2-y==7,y))
 \newcommand{\Bold}{\mathbf{#1}}\left[y = -\sqrt{-x^{2} + 13}, y = \sqrt{-x^{2} + 13}\right] \newcommand{\Bold}{\mathbf{#1}}\left[y = x^{2} - 7\right]
solve([x^2+y^2==13,x^2-y==7],x,y)
 $\newcommand{\Bold}{\mathbf{#1}}\left[\left[x = 3, y = 2\right], \left[x = 2, y = \left(-3\right)\right], \left[x = \left(-2\right), y = \left(-3\right)\right], \left[x = \left(-3\right), y = 2\right]\right]$
plot([-sqrt(13-x^2),sqrt(13-x^2),x^2-7],-sqrt(13),sqrt(13),aspect_ratio=1)  #3) Find all POIs: x^2-y^2==4, y==x^2 solve([x^2-y^2==4,y==x^2],x,y)
 $\newcommand{\Bold}{\mathbf{#1}}\left[\left[x = -\sqrt{\frac{1}{2} i \, \sqrt{15} + \frac{1}{2}}, y = \frac{1}{2} i \, \sqrt{3} \sqrt{5} + \frac{1}{2}\right], \left[x = \sqrt{\frac{1}{2} i \, \sqrt{15} + \frac{1}{2}}, y = \frac{1}{2} i \, \sqrt{3} \sqrt{5} + \frac{1}{2}\right], \left[x = -\sqrt{-\frac{1}{2} i \, \sqrt{15} + \frac{1}{2}}, y = -\frac{1}{2} i \, \sqrt{3} \sqrt{5} + \frac{1}{2}\right], \left[x = \sqrt{-\frac{1}{2} i \, \sqrt{15} + \frac{1}{2}}, y = -\frac{1}{2} i \, \sqrt{3} \sqrt{5} + \frac{1}{2}\right]\right]$
show(solve(x^2-y^2==4,y))
 \newcommand{\Bold}{\mathbf{#1}}\left[y = -\sqrt{x^{2} - 4}, y = \sqrt{x^{2} - 4}\right]
a=plot([-sqrt(x^2-4)],-4,-2,color='red') b=plot([-sqrt(x^2-4)],2,4,color='red') c=plot([sqrt(x^2-4)],-4,-2,color='green') d=plot([sqrt(x^2-4)],2,4,color='green') e=plot([x^2],-4,4,color='orange') show(a+b+c+d+e,aspect_ratio=1)  