Answer to Question #235356 in Linear Algebra for lavanya

Question

Give a geometric description of a single linear equation in three variables. Then give a geometric description of the solution set of a system of 3 linear equations in 3 variables if the system

(a) is inconsistent.

(b) is consistent and has no free variables.

(c) is consistent and has exactly one free variable.

(d) is consistent and has two free variables

Answer 1

If $a, b, c$ and $d$ are real numbers (and if $a, b,$ and $c$ are not all equal to ) then $ax + by + cz = d$ is called a linear equation in three variables.

For three variables, each linear equation determines a plane in three-dimensional space.

(a) Inconsistent system has no solution. Graphically, a system with no solution is represented by three planes with no point in common:

three different planes are parallel,

two planes are identical and the third plane is parallel to them,

two different planes are parallel and the third plane is not parallel to them.

three distinct planes are not parallel and have no point in common.

(b) If the system is consistent and has no free variables, then it has the unique solution.

Solving the system by elimination results in a single ordered triple $(x, y, z).$

Graphically, the ordered triple defines a point that is the intersection of three planes in space.

(c) If the system is consistent and has exactly one free variable, then it has an infinite number of solutions.Graphically, three distinct, non-parallel planes intersect in a line, representing an infinite number of solutions.

(d) If the system is consistent and has two free variables, then it has an infinite number of solutions.Graphically, two identical planes intersect the third one in a line, representing an infinite number of solutions.