Thus the solution set may be a plane, a line, a single point, or the empty set. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set.įor three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. The system has a single unique solution.įor a system involving two variables ( x and y), each linear equation determines a line on the xy- plane.
Ī linear system may behave in any one of three possible ways: The set of all possible solutions is called the solution set. , x n such that each of the equations is satisfied.
The solution set for the equations x − y = −1 and 3 x + y = 9 is the single point (2, 3).Ī solution of a linear system is an assignment of values to the variables x 1, x 2. The number of vectors in a basis for the span is now expressed as the rank of the matrix. A linear system in three variables determines a collection of planes The intersection point is the solution.
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