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In mathematics, simultaneous equations, or systems of equations, are a set of equations containing multiple variables. To solve simultaneous equations, the solver needs to use the provided equations to find the exact value of each variable. The solver may then use a graphical method (by plotting both lines and/or curves in the same graph and finding the exact coordinates of their intersection), the matrix method, the substitution method, and/or the elimination method. Some textbooks refer to the elimination method as the addition method, since it involves adding equations (or constant multiples thereof) to one another, as detailed later in this article.

Consider the following set of equations:

When plotted in a graph, the first equation will appear as a circle, while the second will be a straight line. The graph of an equation is the set of all possible solutions, so the intersections of the graphs of multiple equations represent common solutions to these equations. For equations of the nature of this example (linear equation with a circle), there are three possible solution types:

• The line does not intersect the circle at all. This way, there are no solutions
• The line is a tangent to the circle. This way, there is only one solution (the single point at which the line touches the circle).
• The line cuts the circle. There are two points of intersection, and thus two solutions.

This will be further demonstrated in this article.

Contents 1 Finding solutions 1.1 Substitution method 1.2 Elimination method 1.3 Matrices

[edit] Finding solutions

If there are fewer independent equations than variables, not all variables can be solved for, and so an answer for one variable must be expressed in terms of other variables. In general, to solve a system, one equation is needed for each unknown variable that needs to be solved. If the number of independent equations is the same as the number of variables, then the system should be solvable. Therefore systems are frequently considered where the number of variables and independent equations is the same.

Because of the importance of this, the phrase in the form "E equations, U unknowns" (for example "2 equations 3 unknowns" or "4 equations, 4 unknowns") is often used to describe systems of equations. If E = U or E < U, then usually the system has solutions - usually finite when E = U. If, however, E > U, the system usually has no solutions.

Systems of two equations in two real-valued unknowns are usually one of five different types, having a relationship to the number of solutions:

1. Systems that represent intersecting sets of points such as lines and curves, and that are not of one of the types below. This can be considered the normal type, the others being exceptional in some respect. These systems usually have a finite number of solutions, each formed by the coordinates of one point of intersection.
2. Systems that simplify down to false (for example, equations such as 1 = 0). Such systems have no points of intersection and no solutions. This type is found, for example, when the equations represent parallel lines.
3. Systems in which both equations simplify down to an identity (for example, x = 2x − x and 0y = 0). Any assignment of values to the unknown variables satisfies the equations. Thus, there are an infinite number of solutions: all points of the plane.
4. Systems in which the two equations represent the same set of points: they are mathematically equivalent. (One equation can typically be transformed into the other through algebraic manipulation.) Such systems represent completely overlapping lines, or curves, etc. One of the two equations is redundant and can be discarded. Each point of the set of points corresponds to a solution. Usually, this means there are an infinite number of solutions.
5. Systems in which one (and only one) of the two equations simplifies down to an identity. It is therefore redundant, and can be discarded, like for type 4. Each point of the set of points represented by the other equation is a solution. As for type 4, there are then usually an infinite number of solutions.

The equation x² + y² = 0 can be thought of as the equation of a circle whose radius has shrunk to zero, and so it represents a single point: (x = 0, y = 0), unlike a normal circle containing an infinity of points. This and similar examples show the reason why the types 4 and 5 need the qualification "usually". An example of a system of equations of type 1 with an infinite number of solutions is given by x = |x|, y = |y| (where the notation |•| denotes the absolute value function), whose solutions form a quadrant of the x-y plane. Another example is x = |y|, y = |x|, whose solution represents a ray.

[edit] Substitution method

Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Find an equation that can be solved for one variable, that is, it can be rewritten in the form 〈VARIABLE〉 = 〈EXPRESSION〉, in which the left-hand side variable does not occur in the right-hand side expression. Next, substitute that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved (whether by further application of the substitution method or by other methods), substitute the solutions found for the variables in the above right-hand side expression.

In the above example, we first solve the second equation for x:

and substitute this result into the first equation:

After simplification, this yields the solutions

and by substituting this in x = −2y we obtain the corresponding x values. We now have the two solutions of our system of equations:

[edit] Elimination method

Elimination by judicious multiplication is the other commonly used method to solve simultaneous linear equations. It uses the general principles that each side of an equation still equals the other when both sides are multiplied (or divided) by the same quantity, or when the same quantity is added (or subtracted) from both sides. In multiplication/division, a factor is chosen so that when both sides have equivalent quantities added from another equation in the system (that is, the equations are added), one or more of the variables disappear, the resulting equations are still valid representations in the system, and their smaller number of remaining unknowns thus makes the system of equations easier to solve. As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable. Typically, each "back-substitution" can then allow another variable in the system to be solved.

[edit] Matrices

Systems of equations may also be represented in terms of matrices, allowing various principles of matrix operations to be handily applied to the problem. Systems of simultaneous linear equations are studied in linear algebra and can always be solved; one uses Gaussian elimination or the Cholesky decomposition. To determine approximate solutions to general systems numerically on a computer, the n-dimensional Newton's method may be used. Algebraic geometry is essentially the theory of simultaneous polynomial equations. The question of effective computation with such equations belongs to elimination theory.

Simultaneous equation models are a form of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics.

In modular arithmetic, simple systems of simultaneous congruences can be solved by the method of successive substitution.