Examples for

Equation Solving

Algebraic equations consist of two mathematical quantities, such as polynomials, being equated to each other. Solving equations yields a solution for the independent variables, either symbolic or numeric. In addition to finding solutions to equations, Wolfram|Alpha also plots the equations and their solutions.
Equations

Solve, plot and examine equations with one or more variables.

Solve a linear equation:

4x+3=19

Solve a polynomial equation:

solve x^2 + 4x + 6 = 0

Solve over a specified domain:

solve x^3 - 4x^2 + 6x - 24 = 0 over the reals

Solve an equation with parameters:

solve a x^2 + b x + c = 0 for x

Solve a trigonometric equation:

sin x + cos x = 1
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Systems of Equations

Solve a set of two or more simultaneous equations.

Solve a system of linear equations:

x+y=10, x-y=4

Solve a system of polynomial equations:

x^2+y^2=1, (x-2)^2+(y-1)^2=4
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Systems of Congruences

Find solutions to systems of congruence relations.

Solve a single congruence equation:

solve 5x =2 (mod 3)

Solve systems of congruences:

solve 2x = 10 (mod 12), 3x = 9 (mod 12)

Check if values are equivalent under a given modulus:

17 = 7 mod 10

Solve a congruence involving variables in the modulus:

solve 22 = 10 mod n

Solve systems with each equation under a different modulus:

x = 1 mod 2, x=3 mod 6, x=3 mod 7

Solve multivariate systems of congruences:

x^2 = y^3 mod 2, x=3 mod 7, y=4 mod 7
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Numerical Root Finding

Use numerical approximation methods to solve an equation.

Find a root of an equation using Newton's method:

using Newton's method solve x cos x = 0

Find a root of an equation using the secant method:

using secant method solve x^3-2 at x1=-3 and x2=3

Compute the n th root of a number using the bisection method:

find root of 2 with bisection method
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