Download Jemima`s simultaneous equations

TASK 4
Jemima’s simultaneous equations
Name:
Class:
Date:
Introduction
Jemima has been practising solving simultaneous linear equations by hand. She then uses the
‘solve’ command of her CAS to check that she has found the correct answers. Here is how she
uses the ‘solve’ command, using the equations x + y = 1 and x − y = 3 as an example.
Explorations
a
Solve the following pairs of simultaneous linear equations by hand, and check your answers
using your CAS.
i x − y = 2 and 2x + y = 7
ii 2x − y = 3 and 2x + y = 5
iii x + 3y = 7 and 3x + y = 5
iv 3x + 2y = 1 and 4x − y = 5
Jemima’s teacher Mr Workalot wants to generate some worksheets involving simultaneous linear
equation solving to give his students extra practice. Mr Workalot wants to find pairs of linear
equations that have integer solutions. He is trying to find integer values of a and b that will give
integer solutions to the following pair of linear equations, using a guess and check approach.
x + y = 10 (1)
ax + by = 20 ( 2 )
b Use a guess and check approach to find one pair of integers for a and b that will give an
integer solution for these simultaneous equations.
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CAS-Active Explorations
Jemima suggests to Mr Workalot that he should use the elimination method, in the following
way, to find the solution for y in terms of a and b of the equations above.
■ Multiply equation (1) by a, and label the result as equation (3).
■ Subtract equation (2) from equation (3) and simplify the result to obtain an equation for y
in terms of a and b.
c Use Jemima’s suggestion to obtain an equation for y in terms of a and b. (Show all your
working.)
d Use a similar method to obtain an equation for x in terms of a and b. (Show all your working.)
e Use your CAS as above to check your answers to questions c and d.
[Warning: Enter ax + by = 20 with multiplication signs, i.e. as a × x + b × y = 20, otherwise
your CAS may treat ax and by as single symbols.]
Jemima wonders whether the answers to (1) and (2) for x and y in terms of a and b are always
correct. In particular, what happens if a = b?
f i Suppose a = b = 1. What can you say about the simultaneous equations in this case?
ii The following screenshots show what a CAS reports for the case in part i.
How should you interpret this CAS output?
iii Use your CAS to test the case a = b = –1.
iv Use your CAS to test the case a = b = 2. Can you explain what the output means in this
case?
g Jemima tells Mr Workalot that the original pair of equations will have an integer solution if
a and b are consecutive integers. Explain how she worked this out.
h Jemima tells Mr Workalot that the original pair of equations will have an integer solution if
a and b are just 2 apart. Explain how she worked this out.
i Use your answers for x and y in terms of a and b to help Mr Workalot identify all integer
solutions to the equations for integer values of a and b between –5 and 5 inclusive.
[Hint: One way is to use a spreadsheet to systematically list and calculate the values of x and
y for all the possible integer combinations of a and b in the required range. Alternatively, you
could use your CAS, systematically testing each combination.]
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Permission is granted for this page to be photocopied by the purchasing institution. Copyright © Macmillan Education Australia 2008. Published by Macmillan Education Australia.
16/9/08 11:21:32 AM
Task 4: Jemima’s simultaneous equations
Challenge
Mr Workalot wants some harder simultaneous linear equations for his students to solve. He asks
Jemima to find five combinations of integer values for constants a, b, c and d so that the following
pair of simultaneous linear equations will have integer solutions for x and y.
ax + by = 12
(1)
cx + dy = 30 ( 2 )
He also tells Jemima that the values for the constants must all lie between –5 and 5 inclusive and
that none of the constants can be equal to 1.
Can you help Jemima find five combinations? For each combination, check that the solutions for
x and y really are integers.
Permission is granted for this page to be photocopied by the purchasing institution. Copyright © Macmillan Education Australia 2008. Published by Macmillan Education Australia.
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Teacher notes
This task involves solving simultaneous linear equations. It requires the ability to work and
think in generalisations, and so will be challenging for many Year 9 or Year 10 students.
However, early parts of the task should be accessible to all, provided they have done some
work on the topic of simultaneous linear equations.
The task makes extensive use of a CAS, including as a checking device in the early parts of the
task (which should be completed by hand). It could be helpful to show students in advance
how to use a CAS in the way required in the task. Also, highlight the warning given with
question e. That is, emphasise that when entering an expression such as ax, the multiplication
sign should be used, i.e. a × x.
The critical part of the task involves the solution of a pair of simultaneous linear equations
involving parameters a and b. Students who find the by-hand approach too difficult may still
achieve some success with the use of their CAS.
Different CAS tools auto-simplify in particular ways that may not be immediately obvious, so
it is important that students are able to work with and understand equivalent forms.
− 10(b − 2 )
−(10 b − 20 )
For example, a CAS may express the answer for x as x =
or x =
,
a−b
a−b
10 ( 2 − b)
whereas it has been expressed in the form x =
in the answers.
a−b
The Challenge involves students solving a similar problem without direction. This should only
be attempted by very able students.
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Curriculum links
generalisation
guess, check and improve
integers
nature of solution
parameters
simultaneous equations
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Answers
Explorations
i x = 3, y = 1
ii x = 2, y = 1
iii x = 1, y = 2
iv x = 1, y = −1
b Many possibilities, e.g. a = 1, b = − 1 or a = − 1, b = 1.
a
c
y=
10 ( a − 2 )
d
x =
10 ( 2 − b)
a−b
a−b
e n/a
f i They cannot have a solution, because the first says the sum of x and y is 10 and the second
says the sum of x and y is 20.
ii The equations have no solution. If you draw graphs of the corresponding straight lines, the
lines are parallel, i.e. they do not intersect.
iii There is no solution in this case.
iv The output means that if y is any number, then x is 10 minus this number. In this case, the
second equation is 2x + 2y = 20, which is the same as x + y = 10. So the two equations are
identical, and therefore there are many solutions.
g The solutions must be integers if the denominator a − b in each of the fractional answers for
x and y is 1 or –1. This will happen if a and b are consecutive integers.
h If a and b are just 2 apart, then a − b = 2 or –2. In either case, this number divides exactly into
10, which is in the numerator of each of the fractional answers for x and y. Then the answers
for x and y will be integers.
i There are many combinations of values of a and b that give unique integer solutions. (Refer to
the Solutions.)
Challenge
Many combinations are possible. Refer to the Solutions for some of these and some possible
approaches.
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