Documentation:CHBE Exam Wiki/Unique Solution, No Solution, or Infinite Solutions

Learning Objectives

By the end of this section you should be able to:

1. Understand the difference between unique solutions, no solutions, and infinitely many solutions.
2. Recognize when a matrix has a unique solution, no solutions, or infinitely many solutions.
3. Recognize when a matrix has a unique solution, no solutions, or infinitely many solutions using python.

Introduction

There are three different types of solutions for a system of linear equations: unique solutions, no solutions, or infinitely many solutions. The structure of your matrix after you row reduce will inform you on what type of solution you have.

Unique Solution

The example shown previously in this module had a unique solution. The structure of the row reduced matrix was

${\begin{vmatrix}1&1&-1&|&5\\0&1&-5&|&8\\0&0&1&|&-1\end{vmatrix}}$

and the solution was

$x=1$ $y=3$ $z=-1$

As you can see, each variable in the matrix can have only one possible value, and this is how you know that this matrix has one unique solution.

No solution

Let's suppose you have a system of linear equations that consist of:

$x+y+z=2$ $y-3z=1$ $2x+y+5z=0$

The augmented matrix is

${\begin{vmatrix}1&1&1&|&2\\0&1&-3&|&1\\2&1&5&|&0\end{vmatrix}}$

and the row reduced matrix is

${\begin{vmatrix}1&0&4&|&1\\0&1&-3&|&1\\0&0&0&|&-3\end{vmatrix}}$

As you can see, the final row states that

$0x+0y+0z=-3$

which impossible, 0 cannot equal -3. Therefore this system of linear equations has no solution.

Infinite Solutions

Let's suppose you have a system of linear equations that consist of:

$-3x-5y+36z=10$ $-x+7z=5$ $x+y-10z=-4$

The augmented matrix is

${\begin{vmatrix}-3&-5&36&|&10\\-1&0&7&|&5\\1&1&-10&|&-4\end{vmatrix}}$

and the row reduced matrix is

${\begin{vmatrix}1&0&-7&|&-5\\0&2&-3&|&1\\0&0&0&|&0\end{vmatrix}}$

As you can see, the final row of the row reduced matrix consists of 0. This means that for any value of Z, there will be a unique solution of x and y, therefore this system of linear equations has infinite solutions'--SaidZaidAlkailani (talk) 14:43, 13 January 2018 (PST).