The intersection point is the solution. One method for solving such a system is as follows. If every vector within that span has exactly one expression as a linear systems of equations elimination method pdf of the given left-hand vectors, then any solution is unique.
Thus the solution set may be a plane, a line, a single point, or the empty set. The solution set for two equations in three variables is, in general, a line. In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. Here, “in general” means that a different behavior may occur for specific values of the coefficients of the equations. In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution.
In general, a system with the same number of equations and unknowns has a single unique solution. In general, a system with more equations than unknowns has no solution. The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point.
When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. This is an example of equivalence in a system of linear equations. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point.
It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. Note that any two of these equations have a common solution. The same phenomenon can occur for any number of equations. In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution.