### 2.51.6 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,x−y=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 x2−y2=0,x2−z2=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 [y−z=0,z−x=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]]`