Appendix
A-10: Solving Equations
Example A-10.3
Solve the three equations x2+y−k2=1,y=x2,2 x=−xy−k for x,y,k.
Solution
Maple Solution - Interactive
The Context Panel option Solve≻Solve (explicit) gives the four solutions k=±1,x,y=0,0 and k=5/4,x,y=±3/2,3/4, but the option Solve≻Solve returns the second two solutions in the form
k=54,x=12RootOf_Z2−3,y=34
The RootOf structure is a compressed representation of an algebraic number, that is, a number that is the solution of a polynomial equation. The Context Panel option All Values applies the allvalues command, which converts the RootOf to the algebraic number(s) it represents.
Control-drag the sequence of three equations and press the Enter key.
Context Panel: Solve≻Solve
Context Panel: All Values
x2+y−k2=1,y=x2,2 x=−xy−k
x2+y−k2=1,y=x2,2x=−xy−k
→solve
k=1,x=0,y=0,k=−1,x=0,y=0,k=54,x=12RootOf_Z2−3,y=34
→all values
k=−1,x=0,y=0,k=1,x=0,y=0,k=54,x=123,y=34,k=−1,x=0,y=0,k=1,x=0,y=0,k=54,x=−123,y=34
It might seem that Maple has returned six solutions, not four. However, the solutions for k=±1 are repeated, so there are actually only four distinct solutions.
Maple Solution - Coded
Assign the sequence of equations to eqs.
eqs≔x2+y−k2=1,y=x2,2 x=−xy−k:
Use solve with the Explicit option.
solveeqs,x,y,k,Explicit
If the Explicit option is not used, the RootOf structure is returned, in which case the allvalues command must be applied.
S≔solveeqs,x,y,k:allvaluesS3
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