College Algebra Chapter 6 Matrices and Determinants and Applications Section 6.2 Inconsistent Systems and Dependent Equations

Concepts 1. Identify Inconsistent Systems 2. Solve Systems with Dependent Equations 3. Solve Applications of Systems of Equations

Identify Inconsistent Systems A linear equation in three variables represents a plane in space. The plane x + 2y – 2z = 6 is shown below.

Identify Inconsistent Systems A solution to a system of equations in three variables is a common point of intersection among all the planes in the system.   A system of equations may have no solution. Such a system is said to be inconsistent. An inconsistent system will reduce to a contradiction. This means that the planes do not all intersect.

Example 1: Write a system of linear equations represented by the augmented matrix and solve the system.

Example 2: Write a system of linear equations represented by the augmented matrix and solve the system.

Concepts 1. Identify Inconsistent Systems 2. Solve Systems with Dependent Equations 3. Solve Applications of Systems of Equations

Solve Systems with Dependent Equations A system of linear equations may have infinitely many solutions. In such a case, the equations are said to be dependent. For a system of three equations in three variables, this means that the planes intersect in a common line in space, or the three planes all coincide.

Example 3: Solve the system.

Example 3 continued:

Example 3 continued: This system of three equations and three variables reduces to a system of two equations and three variables. The third equation, 0 = 0, is true regardless of the values of x, y, and z. This system represents two nonparallel planes intersecting in a line. All points on the line are solutions to the system, indicating that there are infinitely many solutions.

Example 3 continued: We can write the solution set in different ways. Since the top two equations both contain the variable y, we can express x and z in terms of y.

Example 3 continued: A system of linear equations that has the same number of equations as variables is called a square system. In (3), we have three equations and three variables in the original system. But we see that the system reduces to a system of two equations and three variables. A system of linear equations cannot have a unique solution unless there are at least the same number of equations as variables.

Example 4: Solve the system.

Example 4 continued:

Example 4 continued: The system is equivalent to a single equation. This system represents 3 coincident planes. The solution set is the set of all ordered triples that satisfy the equation.

Example 5: Solve the system.

Example 5 continued:

Concepts 1. Identify Inconsistent Systems 2. Solve Systems with Dependent Equations 3. Solve Applications of Systems of Equations

Example 6: Consider the network of three one-way streets shown below. x1, x2, and x3 indicate the traffic flow (in vehicles per hour) along the stretches of road AB, AC and CB. The other numbers indicate the traffic flow rates into and out of the intersections A, B, and C.

Example 6 continued: a. Write an equation representing equal flow into and out of intersection A. b. Write an equation representing equal flow into and out of intersection B.

Example 6 continued: c. Write an equation representing equal flow into and out of intersection C. d. Write the system of equations from parts (a) through (c) in standard form.

Example 6 continued: e. Write the reduced row-echelon form of the augmented matrix representing the system of equations from part (d).

Example 6 continued: f. If the flow rate between intersections C and B is 150 vehicles per hour, determine the flow rates x1 and x3.

Example 6 continued: g. If the flow rate between intersections C and B is between 150 and 200 vehicles per hour, inclusive, determine the flow rates x1 and x3.