Solve equations and systems of equations with Wolfram|Alpha
A powerful tool for finding solutions to systems of equations and constraints
Wolfram|Alpha is capable of solving a wide variety of systems of equations. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. Additionally, it can solve systems involving inequalities and more general constraints.
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- Systems of equations »
Tips for entering queries
Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about solving systems of equations.
- solve y = 2x, y = x + 10
- solve system of equations {y = 2x, y = x + 10, 2x = 5y}
- y = x^2 - 2, y = 2 - x^2
- solve 4x - 3y + z = -10, 2x + y + 3z = 0, -x + 2y - 5z = 17
- solve system {x + 2y - z = 4, 2x + y + z = -2, z + 2y + z = 2}
- solve 4 = x^2 + y^2, 4 = (x - 2)^2 + (y - 2)^2
- x^2 + y^2 = 4, y = x
- View more examples »
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What are systems of equations?
A system of equations is a set of one or more equations involving a number of variables.
The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection.
Systems of linear equations are a common and applicable subset of systems of equations. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. The system is said to be inconsistent otherwise, having no solutions. Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent).
More general systems involving nonlinear functions are possible as well. These possess more complicated solution sets involving one, zero, infinite or any number of solutions, but work similarly to linear systems in that their solutions are the points satisfying all equations involved. Going further, more general systems of constraints are possible, such as ones that involve inequalities or have requirements that certain variables be integers.
Solving systems of equations is a very general and important idea, and one that is fundamental in many areas of mathematics, engineering and science.