Polynomials: Equating coefficients 2 (simultaneous equations)

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Ben Brawn

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7 years, 7 months ago

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Ben Brawn 7 years, 7 months ago

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Ben Brawn 7 years, 7 months ago

Created this as a copy of Simultaneous equations.

Name	Status	Author	Last Modified
Simultaneous equations	Ready to use	Ben Brawn	21/02/2025 09:20
Simultaneous equations: linear and quadratic, one point	Ready to use	Ben Brawn	21/02/2025 08:41
Simultaneous equations: linear and quadratic, two points	Ready to use	Ben Brawn	25/02/2025 00:04
Simultaneous equations: simple quadratic and hyperbola, one point	Ready to use	Ben Brawn	21/02/2025 09:13
Simultaneous equations: linear and hyperbola, two points	Ready to use	Ben Brawn	25/02/2025 04:22
Rob's copy of Simultaneous equations: linear and hyperbola, two points	draft	Rob Cade	07/06/2016 11:00
Francis 's copy of Simultaneous equations: linear and quadratic, one point	draft	Francis Duah	12/08/2016 17:58
Preferred Simultaneous equations	draft	Francis Duah	14/09/2016 13:20
Katherine's copy of Simultaneous equations: linear and quadratic, one point	draft	Katherine Tomlinson	02/08/2017 12:48
Simultaneous equations: linear and quadratic, two points	draft	Katherine Tomlinson	02/08/2017 13:50
Polynomials: Equating coefficients 2 (simultaneous equations)	Ready to use	Ben Brawn	14/09/2017 07:56
Polynomials: Equating coefficients 2 (simultaneous equations)	draft	Paul Hancock	24/07/2018 02:35
Polynomials: Equating coefficients 2 (simultaneous equations) [L4 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 10:46
Polynomials: Equating coefficients 2 (simultaneous equations) [L4 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 11:13
cormac's copy of Katherine's copy of Simultaneous equations: linear and quadratic, one point	draft	cormac breen	30/10/2018 15:52
Simultaneous equations	Ready to use	heike hoffmann	14/11/2018 21:12
Intersection: linear and quadratic, two points	Ready to use	heike hoffmann	14/11/2018 22:46
Terry's copy of Simultaneous equations: linear and quadratic, one point	draft	Terry Young	03/04/2019 17:37
Terry's copy of Simultaneous equations: linear and quadratic, two points	draft	Terry Young	03/04/2019 17:37
Simultaneous equations: linear and quadratic, one point	draft	Xiaodan Leng	11/07/2019 06:06
Simultaneous equations: linear and quadratic, two points	draft	Xiaodan Leng	11/07/2019 06:06
Simultaneous equations: simple quadratic and hyperbola, one point	draft	Xiaodan Leng	11/07/2019 06:07
Simultaneous equations: linear and quadratic, two points	draft	Kate Henderson	27/09/2024 10:02

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Question statement

Suppose that during some working, you find that

$\simplify{{-s}x+{-p}={q}*A*x+{r}*B*x+{m}A+{n}B}$

for all $x$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

m	integer	-4
n	integer	-2
p	integer	-7
q	integer	8
r	integer	1
s	integer	9
xcans	rational	-11/12
ycans	rational	-5/3

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-12..12 except 0)

JME function reference

Description

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`

-4

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add

Gap-fill

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

By equating coefficients, determine the values of $A$ and $B$ .

$A=$ Gap 0., $B=$ Gap 1.

Advice

Two polynomials are equal if and only if their corresponding coefficients are also equal.

So we 'equate the coefficients' of the polynomials given in this question, this means that the constant terms must be equal, ie

$\simplify{{-p}={m}*A+{n}*B}\qquad\qquad(1)$

the coefficients of the $x$ terms must be equal, ie

$\simplify{{-s}={q}A+{r}*B}\qquad\qquad(2)$

We now solve these equations simultaneously and find $A=\simplify[fractionnumbers]{{xcans}}$ and $B=\simplify[fractionnumbers]{{ycans}}$ .

The working for simulations equations is given below.

There are many ways to solve these equations simultaneously. Here is one method.

$\simplify{{m}A+{n}B}$	$=$	$\var{-p}$	$(1)$
$\simplify{{q}A+{r}B}$	$=$	$\var{-s}$	$(2)$

Solve one of the equations for one of the variables. Here we solve equation $(1)$ for $B$ :

$\var{n}B$	$=$	$\simplify{{-m}A+{-p}}$

$B$	$=$	$\displaystyle{\simplify{({-m}A+{-p})/({n})}}$	$(3)$

Substitute this expression for $B$ given in $(3)$ into $(2)$ :

$\simplify[all,!collectnumbers]{{q}A+{r}*(({-m}A+{-p})/{n}) =- {s}}$

Collect like terms:

$\simplify[fractionnumbers]{{q-r*m/n}A={-s+r*p/n}}$

Solve for $A$ :

$A=\simplify[fractionnumbers]{{xcans}}$

Now we know the

$A$ value we can determine the corresponding

$B$ value by substituting

$A=\simplify[fractionnumbers]{{xcans}}$ into equation

$(3)$ :

$B$	$=$	$\displaystyle{\simplify[unitdenominator,!collectnumbers,fractionnumbers]{({-m}*({xcans})+{-p})/({n})}}$

	$=$	$\simplify[fractionnumbers]{{ycans}}$

Therefore the values that satisfy equations $(1)$ and $(2)$ are $A=\simplify[fractionnumbers]{{xcans}}$ and $B=\simplify[fractionnumbers]{{ycans}}$ .

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:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is used in the following exams:

Equating coefficients by Ben Brawn in Algebra.
4 Equating coefficients by Ben Brawn in numbas talk 2022.
Graphing and Polynomials by Chris Templet in Chris's workspace.

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Polynomials: Equating coefficients 2 (simultaneous equations)

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Ben Brawn 7 years, 7 months ago

Ben Brawn 7 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

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Error

Polynomials: Equating coefficients 2 (simultaneous equations)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 7 months ago

Ben Brawn 7 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`