No Solution

Definition

The term 'no solution' refers to a situation where a system of equations or inequalities has no values for the variables that satisfy all the equations or inequalities simultaneously. In other words, there is no set of values that can be assigned to the variables that make all the expressions in the system true.

5 Must Know Facts For Your Next Test

1. A system of linear equations in three variables has no solution if the system is inconsistent, meaning the equations are contradictory and cannot be satisfied simultaneously.
2. In a system of nonlinear equations and inequalities in two variables, a no solution scenario can occur when the graphs of the expressions do not intersect at any point.
3. Gaussian elimination, a method for solving systems of linear equations, can result in a no solution outcome if the system is inconsistent and the reduced row echelon form contains a row of all zeros.
4. The presence of a no solution in a system of equations or inequalities indicates that the system is overdetermined, meaning there are more constraints than variables.
5. Identifying a no solution scenario is an important step in understanding the behavior of a system and determining the appropriate solution methods to apply.

Review Questions

• Explain how a system of linear equations in three variables can result in a no solution scenario.
  • A system of linear equations in three variables can have no solution if the system is inconsistent. This means that the equations in the system are contradictory, and there are no values for the three variables that can satisfy all the equations simultaneously. Geometrically, this corresponds to the situation where the three planes represented by the equations do not intersect at a common point. Instead, the planes may be parallel or intersect in a line, resulting in an inconsistent system with no solution.
• Describe the conditions under which a system of nonlinear equations and inequalities in two variables can have no solution.
  • In a system of nonlinear equations and inequalities in two variables, a no solution scenario can occur when the graphs of the expressions do not intersect at any point. This can happen if the graphs of the equations and inequalities are disjoint, meaning they do not share any common points. For example, if one equation represents a circle and the other represents a parabola, and the two graphs do not intersect, then the system has no solution. The absence of a common point of intersection between the graphs indicates that there are no values for the two variables that can satisfy all the expressions in the system simultaneously.
• Analyze how the Gaussian elimination method can lead to a no solution outcome when solving a system of linear equations.
  • When using the Gaussian elimination method to solve a system of linear equations, a no solution scenario can arise if the system is inconsistent. During the row reduction process, the method will ultimately produce a reduced row echelon form of the augmented matrix. If this reduced row echelon form contains a row of all zeros, it indicates that the system is inconsistent and has no solution. This is because the row of all zeros represents an equation that is always false, meaning there are no values for the variables that can satisfy all the equations in the system. The presence of a row of all zeros in the final reduced row echelon form is a clear indicator that the system has no solution.

Related terms

Consistent System: A system of equations or inequalities that has at least one solution that satisfies all the expressions in the system.

Inconsistent System: A system of equations or inequalities that has no solution that satisfies all the expressions in the system.

Infinite Solutions: A system of equations or inequalities that has infinitely many solutions that satisfy all the expressions in the system.