When is solution set empty

We will see in example in Section 2. Another natural question is: are the solution sets for inhomogeneuous equations also spans? As we will see shortly, they are never spans, but they are closely related to spans. The above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc.

The number of free variables is called the dimension of the solution set. We will develop a rigorous definition of dimension in Section 2. Compare with this important note in Section 2. Intuitively, the dimension of a solution set is the number of parameters you need to describe a point in the solution set. For a line only one parameter is needed, and for a plane two parameters are needed. This is similar to how the location of a building on Peachtree Street—which is like a line—is determined by one number and how a street corner in Manhattan—which is like a plane—is specified by two numbers.

In the above example , the solution set was all vectors of the form. We call p a particular solution. It is not hard to see why the key observation is true. But the key observation is true for any solution p.

Again, in the first case the linear operator is invertible while in the other cases it is not. When the operator is not invertible the solution set can be empty, a line in the plane or the plane itself. Each of these equations is the equation of a plane in three-dimensional space. To find solutions to the system of equations, we look for the common intersection of the planes if an intersection exists. Here we have five different possibilities:. Inconsistent systems : All three figures represent three-by-three systems with no solution.

Privacy Policy. Skip to main content. Systems of Equations. Search for:. Systems of Equations in Three Variables. Learning Objectives Solve a system of equations in three variables graphically, using substitution, or using elimination.

Key Takeaways Key Points In a system of equations in three variables , you can have one or more equations, each of which may contain one or more of the three variables, usually x , y , and z. The substitution method involves solving for one of the variables in one of the equations, and plugging that into the rest of the equations to reduce the system. Repeat until there is a single equation left, and then using this equation, go backwards to solve the previous equations.

The graphical method involves graphing the system and finding the single point where the planes intersect. The elimination method involves adding or subtracting multiples of one equation from the other equations, eliminating variables from each of the equations until one variable is left in each equation.

Key Terms system of equations in three variables : A set of one or more equations, each of which may contain one ore more of the three variables usually x, y, and z. Learning Objectives Explain what it means, graphically, for systems of equations in three variables to be inconsistent or dependent, as well as how to recognize algebraically when this is the case.

Key Takeaways Key Points Dependent systems have an infinite number of solutions. Graphically, the infinite number of solutions are on a line or plane that serves as the intersection of three planes in space. Inconsistent systems have no solution. Graphically, a system with no solution is represented by three planes with no point in common. Key Terms Independent system : A system of equations with a single solution.

Dependent system : A system of equations with an infinite number of solutions. For systems of equations in three variables, there are an infinite number of solutions on a line or plane that is the intersection of three planes in space.