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

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Subsection 1.4.1 Class Activities

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Activity 1.4.1.

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

x1+2x2=4+x3=−1

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

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(a)

Let x1=a and write the solution set in the form {[a??]|a∈R}.

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(b)

Let x2=b and write the solution set in the form {[?b?]|b∈R}.

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(c)

Which of these was easier? What features of the RREF matrix [1204001−1] caused this?

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Definition 1.4.2.

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

The pivot columns in an augmented matrix correspond to the bound variables in the system of equations (x1,x3 below). The remaining variables are called free variables (x2 below).

[1204001−1]

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

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Activity 1.4.3.

Find the solution set for the system

2x1−2x2−6x3+x4−x5=3−x1+x2+3x3−x4+2x5=−3x1−2x2−x3+x4+x5=2

by doing the following.

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(a)

Row-reduce its augmented matrix.

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(b)

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

?=a and ?=b.

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(c)

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

?=1+5a+2b,

?=1+2a+3b,

and ?=3+3b.

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(d)

Replace x1 through x5 with the appropriate expressions of a,b in the following set-builder notation.

{[x1x2x3x4x5]|a,b∈R}
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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: ∅ or {} (not 0).

  • Consistent with one solution: e.g. {[123]} (not just [123]).

  • Consistent with infinitely-many solutions: e.g. {[12−3aa]|a∈R} (not just [12−3aa] ).

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Activity 1.4.5.

Show how to find the solution set for the vector equation

x1[101]+x2[01−1]+x3[−15−5]+x4[−313−13]=[−312−12].
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Activity 1.4.6.

Consider the following system of linear equations.

x1−2x3=−35x1+x2−7x3=−185x1−x2−13x3=−12x1+3x2+7x3=−12
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(a)

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

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(b)

Explain how to describe this solution set using set notation.

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Subsection 1.4.2 Videos

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Figure 4. Video: Solving a system of linear equations with infinitely-many solutions
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Subsection 1.4.3 Slideshow

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

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Exercises 1.4.4 Exercises

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

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Subsection 1.4.5 Mathematical Writing Explorations

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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.

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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.

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Subsection 1.4.6 Sample Problem and Solution

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Sample problem Example B.1.4.

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