## Section 1.3 Counting Solutions for Linear Systems (LE3)

### Learning Outcomes

Determine the number of solutions for a system of linear equations or a vector equation.

### Subsection 1.3.1 Class Activities

#### Activity 1.3.1.

Free browser-based technologies for mathematical computation are available online.

In the dropdown on the right, you can select a number of different languages. Select "Octave" for the Matlab-compatible syntax used by this text.

Type

`rref([1,3,2;2,5,7])`

and then press the`Evaluate`button to compute the \(\RREF\) of \(\left[\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 5 & 7 \end{array}\right]\text{.}\)

Since the vertical bar in an augmented matrix does not affect row operations, the \(\RREF\) of \(\left[\begin{array}{cc|c} 1 & 3 & 2 \\ 2 & 5 & 7 \end{array}\right]\) may be computed in the same way.

#### Activity 1.3.2.

In the HTML version of this text, code cells are often embedded for your convenience when RREFs need to be computed.

Try this out to compute \(\RREF\left[\begin{array}{cc|c} 2 & 3 & 1 \\ 3 & 0 & 6 \end{array}\right]\text{.}\)

#### Activity 1.3.3.

Consider the following system of equations.

##### (a)

Convert this to an augmented matrix and use technology to compute its reduced row echelon form:

##### (b)

Use the \(\RREF\) matrix to write a linear system equivalent to the original system.

##### (c)

How many solutions must this system have?

Zero

Only one

Infinitely-many

#### Activity 1.3.4.

Consider the vector equation

##### (a)

Convert this to an augmented matrix and use technology to compute its reduced row echelon form:

##### (b)

Use the \(\RREF\) matrix to write a linear system equivalent to the original system.

##### (c)

How many solutions must this system have?

Zero

Only one

Infinitely-many

#### Activity 1.3.5.

Is \(0=1\) the only possible logical contradiction obtained from the RREF of an augmented matrix?

Yes, \(0=1\) is the only possible contradiction from an RREF matrix.

No, \(0=17\) is another possible contradiction from an RREF matrix.

No, \(x=0\) is another possible contradiction from an RREF matrix.

No, \(x=y\) is another possible contradiction from an RREF matrix.

#### Activity 1.3.6.

Consider the following linear system.

##### (a)

Find its corresponding augmented matrix \(A\) and find \(\RREF(A)\text{.}\)

##### (b)

Use the \(\RREF\) matrix to write a linear system equivalent to the original system.

##### (c)

How many solutions must this system have?

Zero

One

Infinitely-many

#### Fact 1.3.7.

By finding \(\RREF(A)\) from a linear system's corresponding augmented matrix \(A\text{,}\) we can immediately tell how many solutions the system has.

If the linear system given by \(\RREF(A)\) includes the contradiction \(0=1\text{,}\) that is, the row \(\left[\begin{array}{ccc|c}0&\cdots&0&1\end{array}\right]\text{,}\) then the system is

*inconsistent*, which means it has*zero*solutions and its solution set is written as \(\emptyset\) or \(\{\}\text{.}\)If the linear system given by \(\RREF(A)\) sets each variable of the system to a single value; that is, \(x_1=s_1\text{,}\) \(x_2=s_2\text{,}\) and so on; then the system is

*consistent*with exactly*one*solution \(\left[\begin{array}{c}s_1\\s_2\\\vdots\end{array}\right]\text{,}\) and its solution set is \(\setList{ \left[\begin{array}{c}s_1\\s_2\\\vdots\end{array}\right] }\text{.}\)Otherwise, the system must be

*consistent*with*infinitely-many*different solutions. We'll learn how to find such solution sets in SectionÂ 1.4.

#### Activity 1.3.8.

For each vector equation, write an explanation for whether each solution set has no solutions, one solution, or infinitely-many solutions. If the set is finite, describe it using set notation.

##### (a)

##### (b)

##### (c)

### Subsection 1.3.2 Videos

### Subsection 1.3.3 Slideshow

Slideshow of activities available at `https://teambasedinquirylearning.github.io/linear-algebra/2022/LE3.slides.html`

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### Exercises 1.3.4 Exercises

Exercises available at `https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/LE3/`

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### Subsection 1.3.5 Mathematical Writing Explorations

#### Exploration 1.3.9.

A system of equations with all constants equal to 0 is called homogeneous. These are addressed in detail in section SectionÂ 2.9

Choose three systems of equations from this chapter that you have already solved. Replace the constants with 0 to make the systems homogeneous. Solve the homogeneous systems and make a conjecture about the relationship between the earlier solutions you found and the associated homogeneous systems.

Prove or disprove. A system of linear equations is homogeneous if an only if it has the the zero vector as a solution.

### Subsection 1.3.6 Sample Problem and Solution

Sample problem ExampleÂ B.1.3.