Chapter 16.6, Problem 13OE

To determine

To Explain: If the system of equations does not have a unique solution, does it mean that system has no solution?

Answer to Problem 13OE

No, it means that there can be either no solution or infinitely many solutions can be there.

Explanation of Solution

Given:The system of equations does not have a unique solution.

If given system of equations are:

  $\begin{array}{l}{a}_{{}^1}x+b_{{}^1}y=c_1\\ a_{{}^2}x+b_{{}^2}y=c_2\end{array}$

The above system can be represented as:

  $\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}{c}_1\\ c_2\end{array}\right]$

  $AX=C$

  $X=A^{-1}C$

The solution to the system of equations is the matrix $X$ .

There exists a unique solution to the system of equations if the matrix $A$ is invertible.

For checking whether a matrix is invertible it is needed to verify that the determinant of the matrix $A$ is non-zero or not.

If the determinant of the given matrix is non-zero then the matrix is invertible.

And there exists a unique solution for the system of equations.

If the determinant of the matrix $A$ is zero, then there can be one of the possibilities:

1. Either the system of equations has no solution possible i.e. parallel lines.
2. Or the system of equations has infinitely many solutions i.e. identical lines.