Simultaneous Equations - Finding Solutions

Finding Solutions

Sometimes not all variables can be solved for, and so an answer for at least one variable must be expressed in terms of other variables and so the set of all solutions is infinite; this is typical for the case where the system has fewer equations than variables. If the number of equations is the same as the number of variables, then probably (but not necessarily) the system is exactly solvable in the sense that the set of its solutions is finite; for a system of linear equations in this case there is exactly one solution, for other systems to have several solutions is also typical. A consistent system is a system of equations with at least one solution. Sometimes a system is inconsistent, or has no solution; this is typical for the case where the system has more equations than variables. If these rules about connection between number of solutions and numbers of equations and variables do not hold, then such situation is often referred to as dependence between equations or between their left parts. For instance, this occurs in linear systems if one equation is a simple multiple of the other (representing the same line, e.g. 2x + y = 3 and 4x + 2y = 6) or if the ratio of like variables in two linear equations is the same (representing parallel lines, e.g. 2x + y = 3 and 6x + 3y = 7 where the ratio of comparable letters is 3).

Systems of two equations in two real-value unknowns usually appear as 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, as per the previous type. Each point of the set of points represented by the other equation is a solution of which there are then usually an infinite number.

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 last two types described above need the qualification "usually". An example of a system of equations of the first type described above 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. Another example is (x+1)(x+y)=0, (y+1)(x+y)=0, whose solution represents a line and a point.