Gaussian elimination to solve a system of linear equations - scaffolded

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From users who are members of Linear Algebra 1st year :

Tatiana Tyukina	said	Ready to use	4 years, 8 months ago
Julia Goedecke	said	Needs to be tested	4 years, 8 months ago

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Julia Goedecke 4 years, 8 months ago

Published this.

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Tatiana Tyukina 4 years, 8 months ago

Gave some feedback: Ready to use

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Julia Goedecke commented 4 years, 8 months ago

Marks for each part finalised.

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Julia Goedecke commented 4 years, 8 months ago

Still need to adapt the marks per parts to make them a bit more round. Will do that once someone has tested that it works.

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Julia Goedecke 4 years, 8 months ago

Saved a checkpoint:

Adapted the "introductory" Gaussian elimination question to include two cases: sometimes this one has a unique solution, and sometimes it has a free variable.

Restore this checkpoint

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Julia Goedecke 4 years, 8 months ago

Gave some feedback: Needs to be tested

○

Julia Goedecke 4 years, 8 months ago

Created this as a copy of Gaussian elimination to solve a system of linear equations - Introductory.

Name	Status	Author	Last Modified
Gaussian elimination to solve a system of linear equations - Introductory	Ready to use	Julia Goedecke	25/08/2021 12:57
Gaussian elimination to solve a system of linear equations - scaffolded	Ready to use	Julia Goedecke	25/08/2021 12:57
Gaussian elimination to solve a system of equations	Ready to use	Andrew Brown	12/09/2022 14:48
Gaussian elimination to solve a system of linear equations	Ready to use	Andrew Brown	12/09/2022 14:48

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Question statement

Solve the system of equations using Gaussian Elimination:

$\begin{eqnarray*} &x&+\;&\var{b}y&+\;\var{b*a-b}z&=&\var{c3} \\ &\var{a}x&+\;&\var{a*b-1}y&+\;\var{a^2*b-a-a*b}z&=&\var{c2}\\ &\var{a*c}x&+\;&\var{c*b}y&+\simplify[!zeroFactor,unitFactor]{{d}*z}&=&\var{c1} \end{eqnarray*}$
Part a) Represent this as a system of equations using a matrix.
Part b) Introduce zeros in the first column using the first row.
Part c) Introduce zeros in the second column below the second entry in the second row using the second row.
Part d) Determine whether the system has no solutions, a unique solution or infinitely many solutions.
Part e) If the system has solutions, calculate the reduced row echelon form of the matrix.
Part f) Read off the general solution from the reduced row echelon form, introducing free variables if necessary.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
c	integer	1
b	integer	5
c3	integer	3
c2	integer	1
y	number	11
x	number	-52
c1	integer	-153
z	number	0
d	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..6)

JME function reference

Description

Note that a cannot be 0 or 1, which makes sure the steps of the Gauss algorithm come out as expected and don't skip one.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

Statement
Advice
"Part a)" → "Gap 3" - Minimum accepted value
"Part a)" → "Gap 3" - Maximum accepted value
"Part a)" → "Gap 5" - Minimum accepted value
"Part a)" → "Gap 5" - Maximum accepted value
"Part a)" → "Gap 6" - Minimum accepted value
"Part a)" → "Gap 6" - Maximum accepted value
"Part a)" → "Gap 7" - Minimum accepted value
"Part a)" → "Gap 7" - Maximum accepted value
"Part a)" → "Gap 9" - Minimum accepted value
"Part a)" → "Gap 9" - Maximum accepted value
"Part b)" - prompt
"Part b)" → "Gap 4." - Minimum accepted value
"Part b)" → "Gap 4." - Maximum accepted value
"Part b)" → "Gap 5." - Minimum accepted value
"Part b)" → "Gap 5." - Maximum accepted value
"Part b)" → "Gap 7." - Minimum accepted value
"Part b)" → "Gap 7." - Maximum accepted value
"Part c)" - prompt
"Part c)" → "Gap 1." - Minimum accepted value
"Part c)" → "Gap 1." - Maximum accepted value
"Part c)" → "Gap 2." - Minimum accepted value
"Part c)" → "Gap 2." - Maximum accepted value
"Part c)" → "Gap 4." - Minimum accepted value
"Part c)" → "Gap 4." - Maximum accepted value
"Part e)" → "Gap 3." - Minimum accepted value
"Part e)" → "Gap 3." - Maximum accepted value
"Part e)" → "Gap 6." - Minimum accepted value
"Part e)" → "Gap 6." - Maximum accepted value
"Part f)" → "Gap 1." - Correct answer

Parts

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Part f) Gap-fill

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Write down the entries of the matrix you will use for Gaussian Elimination, remember to include the constants as the last column.

$\left( \begin{matrix} \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{,}\end{matrix} \right.$	Gap 1	Gap 2	Gap 3	$\left. \begin{matrix} \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\end{matrix} \middle\| \begin{matrix} \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\end{matrix} \right.$	Gap 4	$\left) \begin{matrix} \phantom{.} \\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.} \end{matrix} \right.$
	Gap 5	Gap 6	Gap 7		Gap 8
	Gap 9	Gap 10	Gap 11		Gap 12

Advice

For parts a) to c), look at the revealed answers for this question. All the information needed is there.

Part d) We have $3$ leading $1$ s in the matrix in part c), one in each row and each column. Therefore we have a unique solution. As the third row is completely zero, we have no leading $1$ in the third column, and $z$ can be any value, so we have a free variable. This means we have infinitely many solutions.

Part e) Since we have a leading $1$ in the second entry in the second row, we can clear the second entry in the first row. Since the second entry in the first row is $\var{b}\neq 0$ , the matrix is definitely not yet in reduced row echelon form.
For the rest of this part, look at the revealed answers.

Part f) As the third row is a zero row, we get $z=t$ as a free variable: $z$ is not restricted at all. This then also appears in the general solutions for $x$ and $y$ : $x-\var{b}z=\var{x}$ , so $x=\var{x}+\var{b}t$ , and $y+\var{a}z=\var{a*c3-c2}$ , so $y=\var{a*c3-c2}-\var{a}t$ .
You can read off the unique solution from the final matrix.

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

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Gaussian elimination to solve a system of linear equations - scaffolded

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Julia Goedecke 4 years, 8 months ago

Tatiana Tyukina 4 years, 8 months ago

Julia Goedecke commented 4 years, 8 months ago

Julia Goedecke commented 4 years, 8 months ago

Julia Goedecke 4 years, 8 months ago

Julia Goedecke 4 years, 8 months ago

Julia Goedecke 4 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

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Error

Gaussian elimination to solve a system of linear equations - scaffolded

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julia Goedecke 4 years, 8 months ago

Tatiana Tyukina 4 years, 8 months ago

Julia Goedecke commented 4 years, 8 months ago

Julia Goedecke commented 4 years, 8 months ago

Julia Goedecke 4 years, 8 months ago

Julia Goedecke 4 years, 8 months ago

Julia Goedecke 4 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`