### 5.55.6  Solving equation(s): solve

solve solves an equation or a system of polynomial equations. It takes 2 arguments:

• Solving an equation
solve takes as arguments an equation between two expressions or an expression (=0 is omitted), and a variable name (by default x).
solve solves this equation.
• Solving a system of polynomial equations
solve takes as arguments two vectors : a vector of polynomial equations and a vector of variable names.
solve solves this polynomial equation system.

Remarks:

• In real mode, solve returns only real solutions. To have the complex solutions, switch to complex mode, e.g. by checking Complex in the cas configuration, or use the cSolve command.
• For trigonometric equations, solve returns by default the principal solutions. To have all the solutions check All_trig_sol in the cas configuration.

Examples :

• Solve x4−1=3
Input :
solve(x`^`4-1=3)
Output in real mode :
[sqrt(2),-(sqrt(2))]
Output in complex mode :
[sqrt(2),-(sqrt(2)),(i)*sqrt(2),-((i)*sqrt(2))]
• Solve exp(x)=2
Input :
solve(exp(x)=2)
Output in real mode :
[log(2)]
• Find x,y such that x+y=1,xy=0
Input :
solve([x+y=1,x-y],[x,y])
Output :
[[1/2,1/2]]
• Find x,y such that x2+y=2,x+y2=2
Input :
solve([x`^`2+y=2,x+y`^`2=2],[x,y])
Output :
[[-2,-2],[1,1],[(-sqrt(5)+1)/2,(1+sqrt(5))/2],
[(sqrt(5)+1)/2,(1-sqrt(5))/2]]
• Find x,y,z such that x2y2=0,x2z2=0
Input :
solve([x`^`2-y`^`2=0,x`^`2-z`^`2=0],[x,y,z])
Output :
[[x,x,x],[x,-x,-x],[x,-x,x],[x,x,-x]]
• Solve cos(2*x)=1/2
Input :
solve(cos(2*x)=1/2)
Output :
[pi/6,(-pi)/6]
Output with All_trig_sol checked :
[(6*pi*n_0+pi)/6,(6*pi*n_0-pi)/6]
• Find the intersection of a straight line (given by a list of equations) and a plane.
For example, let D be the straight line of cartesian equations [yz=0,zx=0] and let P the plane of equation x−1+y+z=0. Find the intersection of D and P.
Input :
solve([[y-z=0,z-x=0],x-1+y+z=0],[x,y,z])
Output :
[[1/3,1/3,1/3]]