## Section 1.4 Linear Systems with Infinitely-Many Solutions (LE4)

### Learning Outcomes

Compute the solution set for a system of linear equations or a vector equation with infinitly many solutions.

### Subsection 1.4.1 Class Activities

#### Activity 1.4.1.

Consider this simplified linear system found to be equivalent to the system from Activity 1.3.6:

Earlier, we determined this system has infinitely-many solutions.

##### (a)

Let \(x_1=a\) and write the solution set in the form \(\setBuilder { \left[\begin{array}{c} a \\ \unknown \\ \unknown \end{array}\right] }{ a \in \IR } \text{.}\)

##### (b)

Let \(x_2=b\) and write the solution set in the form \(\setBuilder { \left[\begin{array}{c} \unknown \\ b \\ \unknown \end{array}\right] }{ b \in \IR } \text{.}\)

##### (c)

Which of these was easier? What features of the RREF matrix \(\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 0 & 4 \\ 0 & 0 & \circledNumber{1} & -1 \end{array}\right]\) caused this?

#### Definition 1.4.2.

Recall that the pivots of a matrix in \(\RREF\) form are the leading \(1\)s in each non-zero row.

The pivot columns in an augmented matrix correspond to the bound variables in the system of equations (\(x_1,x_3\) below). The remaining variables are called free variables (\(x_2\) below).

To efficiently solve a system in RREF form, assign letters to the free variables, and then solve for the bound variables.

#### Activity 1.4.3.

Find the solution set for the system

by doing the following.

##### (a)

Row-reduce its augmented matrix.

##### (b)

Assign letters to the free variables (given by the non-pivot columns):

\(\unknown = a\) and \(\unknown = b\text{.}\)

##### (c)

Solve for the bound variables (given by the pivot columns) to show that

\(\unknown = 1+5a+2b\text{,}\)

\(\unknown = 1+2a+3b\text{,}\)

and \(\unknown=3+3b\text{.}\)

##### (d)

Replace \(x_1\) through \(x_5\) with the appropriate expressions of \(a,b\) in the following set-builder notation.

#### Remark 1.4.4.

Don't forget to correctly express the solution set of a linear system. Systems with zero or one solutions may be written by listing their elements, while systems with infinitely-many solutions may be written using set-builder notation.

*Inconsistent*: \(\emptyset\) or \(\{\}\) (not \(0\)).*Consistent*with*one solution*: e.g. \(\setList{ \left[\begin{array}{c}1\\2\\3\end{array}\right] }\) (not just \(\left[\begin{array}{c}1\\2\\3\end{array}\right]\)).*Consistent*with*infinitely-many solutions*: e.g. \(\setBuilder { \left[\begin{array}{c}1\\2-3a\\a\end{array}\right] }{ a\in\IR }\) (not just \(\left[\begin{array}{c}1\\2-3a\\a\end{array}\right]\) ).

#### Activity 1.4.5.

Show how to find the solution set for the vector equation

#### Activity 1.4.6.

Consider the following system of linear equations.

##### (a)

Explain how to find a simpler system or vector equation that has the same solution set.

##### (b)

Explain how to describe this solution set using set notation.

### Subsection 1.4.2 Videos

### Subsection 1.4.3 Slideshow

Slideshow of activities available at `https://teambasedinquirylearning.github.io/linear-algebra/2022/LE4.slides.html`

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### Exercises 1.4.4 Exercises

Exercises available at `https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/LE4/`

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### Subsection 1.4.5 Mathematical Writing Explorations

#### Exploration 1.4.7.

Construct a system of 3 equations in 3 variables having:

0 free variables

1 free variable

2 free variables

In each case, solve the system you have created. Conjecture a relationship between the number of free variables and the type of solution set that can be obtained from a given system.

#### Exploration 1.4.8.

For each of the following, decide if it's true or false. If you think it's true, can we construct a proof? If you think it's false, can we find a counterexample?

If the coefficient matrix of a system of linear equations has a pivot in the rightmost column, then the system is inconsistent.

If a system of equations has two equations and four unknowns, then it must be consistent.

If a system of equations having four equations and three unknowns is consistent, then the solution is unique.

Suppose that a linear system has four equations and four unknowns and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.

Suppose that a linear system has five equations and three unknowns and that the coefficient matrix has a pivot in every column. Then the linear system is consistent and has a unique solution.

### Subsection 1.4.6 Sample Problem and Solution

Sample problem Example B.1.4.