Chapter 5, Systems of Equations and Inequalities - Section 5.4 - Systems of Nonlinear Equations - 5.4 Exercises - Page 466: 20

$(0,0)$ and $(1,1)$

Step 1. Rewrite the equations. $x=y^2$ $y=x^2$ Step 2. Substitute the first equation to the second one. $y=(y^2)^2$ $y=y^4$ $y^4-y=0$ $y(y^3-1)=0$ $y(y-1)(y^2+y+1)=0$ (Note that $y^2+y+1>0$) $y(y-1)=0$ $y=0$ and $y=1$ Step 3. Solve for $x$. For $y=0$, $x=0^2=0$ For $y=1$, $x=1^2=1$ Step 4. Conclude the solutions. The solutions are $(0,0)$ and $(1,1)$.