Solve Command

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Note: Commands Solve and Solutions solve an equation or a system of equations over the real numbers symbolically. To solve equations numerically, use the NSolve Command. For solving equations in complex numbers see CSolve Command.

CAS Syntax

Solve[ <Equation in x> ]
Solves a given equation for the main variable and returns a list of all solutions.
Example:
Solve[x^2 = 4x] yields {x = 4, x = 0}, the solutions of x2 = 4x.
Solve[ <Equation>, <Variable> ]
Solves an equation for a given unknown variable and returns a list of all solutions.
Example:
Solve[x * a^2 = 4a, a] yields {a = \frac{4}{x}, a = 0}, the solutions of xa2 = 4a.
Solve[ <List of Equations>, <List of Variables> ]
Solves a set of equations for a given set of unknown variables and returns a list of all solutions.
Examples:
  • Solve[{x = 4 x + y , y + x = 2}, {x, y}] yields ( x = -1, y = 3 ), the sole solution of x = 4x + y and y + x = 2
  • Solve[{2a^2 + 5a + 3 = b, a + b = 3}, {a, b}] yields {{a = 0, b = 3}, {a = -3, b = 6}}.
Solve[ <Equation>, <Variable> , <List of assumptions>]
Solves an equation for a given unknown variable with the list of assumptions and returns a list of all solutions.
Examples:
  • Solve[u *x < a,x, u>0] yields {x < a / u}, the sole solution of u *x < a assuming that u>0
  • Solve[u *x < a,x, {u<0, a<0}] yields {x > a / u}.
Note:
  • The right hand side of equations (in any of the above syntaxes) can be omitted. If the right hand side is missing, it is treated as 0.
  • Sometimes you need to do some manipulation to allow the automatic solver to work, for example Solve[TrigExpand[sin(5/4 π + x) - cos(x - 3/4 π) = sqrt(6) * cos(x) - sqrt(2)]] .
  • For piecewise-defined functions, you will need to use NSolve
Solve[ <List of Parametric Equations>, <List of Variables> ]
Solves a set of parametric equations for a given set of unknown variables and returns a list of all solutions.
Example:
  • Solve[{(x, y) = (3, 2) + t*(5, 1), (x, y) = (4, 1) + s*(1, -1)}, {x, y, t, s}] yields {{x = 3, y = 2, t = 0, s = -1}}.
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