Chapter 10, Problem 7RCC

(a)

To determine

To describe: an inconsistent system

Answer to Problem 7RCC

An inconsistent linear system is one that has no solutions

Explanation of Solution

An inconsistent linear system is one that has no solutions. If, when Gaussian elimination to the system is applied, a false equation (such as 0=1) is arrived, then the system has no solutions. No values assigned to the variables can ever make a false statement true, so there exists no combination of values that solve the system, and there is no solution.

If there is a system with three equations and three variables, an instance of no solution means that there is no point that is shared by all three planes formed by the equations.

This is easier to spot if two equations that are almost identical is arrived, but have one value of difference. For example, in the book's example, the system

  $\{\begin{array}{l}x+2y-2z=1\text{ Equation 1}\\ 2x+\text{2}y-z\text{ =6 Equation 2}\\ 3x+4y-3z=5\text{ Equation 3}\end{array}$

is manipulated using Gaussian elimination, and becomes

  $\{\begin{array}{l}x+2y-2z=1\text{ Equation 1}\\ -\text{2}y-3z\text{ =4 Equation 2}\\ -2y+3z=2\text{ Equation 3}\end{array}$

Now, if the second and third equations are subtracted, false statement is arrived.

  $\begin{array}{l}-2y+3z=4\text{ Equation 2}\\ -[-\text{ 2}y\text{ +3z}=2]\left(-1\right)\times \text{ Equation 3}\\ 0=2\text{ Equation 2+}\left(-1\right)\text{+Equation 3}=\text{new Equation 3}\end{array}$

Since $\text{0}\ne \text{2}$ , false statement is arrived. As the system has no solution, it is inconsistent

(b)

To determine

To describe: dependent system

Answer to Problem 7RCC

A dependent linear system is one that has infinitely many solutions.

Explanation of Solution

A dependent linear system is one that has infinitely many solutions. If, when Gaussian elimination to the system is applied, a true equation (such as $0=0$ ) is arrived, then that equation can be dropped from the system. With three variables but only two equations, there will be no numerical solution

write two of the variables (say, x and y) in terms of the third (z). However, since the third can take on any value, the system has infinitely many solutions. This is called as a dependent system. For every value of z, there is a solution point of the form (x, y, z) that depends on z.

This is easier to spot if two equations that are identical is arrived, or are multiples of each other. For example, in the book's example, the system

  $\{\begin{array}{l}x-y+5z=-2\\ 2x+y+4z\text{ = 2 }\\ 2x+4y-2z=8\end{array}$

is manipulated using Gaussian elimination, and becomes

  $\{\begin{array}{l}x-y+5z=-2\\ 3y-6z\text{ = 6 }\\ \text{ 6}y-12z=12\end{array}$

Since Equations 2 and 3 are multiples of each other, subtract two of Equation 2 from Equation 3 and get

  $\{\begin{array}{l}x-y+5z=-2\\ 3y-6z\text{ = 6 }\\ 0=0\end{array}$

The third equation can be dropped, since it is true, but provides no extra information about the variables.

  $\{\begin{array}{l}x-y+5z=-2\\ 3y-6z\text{ = 6 }\end{array}$

Then, x and y are written in term of z, and a dummy variable t serves are the argument of the solution. The solution (in (x, y, z) form) is $\left(-3t,2t+2,t\right)$ . Since there are infinitely many solutions, the system is dependent.