INTERMEDIATE 2 ADDITIONAL QUESTION BANK

1
INTERMEDIATE 2 ADDITIONAL QUESTION BANK
You have chosen to study
Simultaneous Equations
UNIT 2
Please choose a question to attempt from the
following
1
2
3
4
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2
Simultaneous Equations Question 1
The diagram shows the graph of 3x 2y 17.
Copy the diagram and on your diagram draw the
graph of y 2x 2 , hence solve 3x 2y
17 y 2x
2
3x 2y 17.
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3
Simultaneous Equations Question 1
The diagram shows the graph of 3x 2y 17.
Copy the diagram and on your diagram draw the
graph of y 2x 2 , hence solve 3x 2y
17 y 2x
2
To draw graph Construct a table of values with
at least 2 x-coordinates.
Plot and join points. Solution is where lines
cross
3x 2y 17.
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4
Simultaneous Equations Question 1
The diagram shows the graph of 3x 2y 17.
Copy the diagram and on your diagram draw the
graph of y 2x 2 , hence solve 3x 2y
17 y 2x
2
Solution is x 3 y 4
3x 2y 17.
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5
Question 1
1. Construct a table of values with at least 2 x
coordinates.
The diagram shows the graph of 3x 2y 17.
y 2x - 2
x 0 5
Copy the diagram and on your diagram draw the
graph of y 2x 2 , hence solve 3x 2y
17 y 2x
2
y -2 8
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Question 1
2. Plot and join points. Solution is where lines
cross.
The diagram shows the graph of 3x 2y 17.
Copy the diagram and on your diagram draw the
graph of y 2x 2 , hence solve 3x 2y
17 y 2x
2
Begin Solution
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Solution is x 3 y 4
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Markers Comments
1. Construct a table of values with at least 2 x
coordinates.
y 2x - 2
x 0 5
y -2 8
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8
Markers Comments
2. Plot and join points. Solution is where lines
cross.
Plot C(0, -1) and draw line with m 2
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Solution is x 3 y 4
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9
Simultaneous Equations Question 1B
The diagram shows the graph of 2x y 8.
Copy the diagram and on your diagram draw the
graph of y 1/2x 3 , hence solve 2x y
8 y 1/2x 3
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10
Simultaneous Equations Question 1B
The diagram shows the graph of 2x y 8.
Copy the diagram and on your diagram draw the
graph of y 1/2x 3 , hence solve 2x y
8 y 1/2x 3
To draw graph Construct a table of values with
at least 2 x-coordinates.
Plot and join points. Solution is where lines
cross
What would you like to do now?
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Reveal answer only
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11
Simultaneous Equations Question 1B
The diagram shows the graph of 2x y 8.
Copy the diagram and on your diagram draw the
graph of y 1/2x 3 , hence solve 2x y
8 y 1/2x 3
What would you like to do now?
Solution is x 2 y 4
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12
Question 1B
1. Construct a table of values with at least 2 x
coordinates.
The diagram shows the graph of 2x y 8.
y 1/2x 3
x 0 6
y 3 6
Copy the diagram and on your diagram draw the
graph of y 1/2x 3 , hence solve 2x y
8 y 1/2x 3
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Question 1B
2. Plot and join points. Solution is where lines
cross.
The diagram shows the graph of 2x y 8.
y 1/2x 3
x 0 6
y 3 6
Copy the diagram and on your diagram draw the
graph of y 1/2x 3 , hence solve 2x y
8 y 1/2x 3
What would you like to do now?
Begin Solution
Continue Solution
Comments
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Solution is x 2 y 4
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Markers Comments
1. Construct a table of values with at least 2 x
coordinates.
y 1/2x 3
x 0 6
y 3 6
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15
Markers Comments
2. Plot and join points. Solution is where lines
cross.
y 1/2x 3
x 0 6
y 3 6
Plot C(0, 3) and draw line with m ½
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Solution is x 2 y 4
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16
Simultaneous Equations Question 2
Solve 3u - 2v 4 2u 5v 9
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17
Simultaneous Equations Question 2
Eliminate either variable by making coefficient
same.
Solve 3u - 2v 4 2u 5v 9
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Substitute found value into either of original
equations.
Reveal answer only
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18
Simultaneous Equations Question 2
Solve 3u - 2v 4 2u 5v 9
Solution is u 2 v 1
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19
Question 2
1. Eliminate either us or vs by making
coefficient same.
Solve 3u - 2v 4 2u 5v 9
1
2
Now get
(x5)
1

3
(x2)
2
4

3
4

Begin Solution
Continue Solution
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Question 2
2. Substitute found value into either of original
equations.
Solve 3u - 2v 4 2u 5v 9
1
2
Substitute 2 for u in equation
2
4 5v 9
5v 5
v 1
Solution is u 2 v 1
Begin Solution
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21
Markers Comments
Note When the signs are the same subtract
to eliminate. When the signs are different
add to eliminate.
1. Eliminate either us or vs by making
coefficient same.
1
2
Now get
(x5)
1

3
(x2)
2
4

3
4

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22
Markers Comments
Note When the signs are the same subtract
to eliminate. When the signs are different
add to eliminate.
1. Eliminate either us or vs by making
coefficient same.
1
2
Now get
(x5)

3
1
(x2)
2
4

3
4

Next Comment
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23
Simultaneous Equations Question 2B
Solve 5p 3q 0 4p 5q -2.6
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24
Simultaneous Equations Question 2B
Eliminate either variable by making coefficient
same.
Solve 5p 3q 0 4p 5q -2.6
What would you like to do now?
Substitute found value into either of original
equations.
Reveal answer only
Go to full solution
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25
Simultaneous Equations Question 2B
Solve 5p 3q 0 4p 5q -2.6
Solution is q -1 q 0.6
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26
Question 2B
1. Eliminate either ps or qs by making
coefficient same.
Solve 5p 3q 0 4p 5q -2.6
1
2
Now get
(x4)
1

3
(x5)
2
4

4
3
-
Begin Solution
Continue Solution
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Question 2B
2. Substitute found value into either of original
equations.
Solve 5p 3q 0 4p 5q -2.6
1
2
Substitute -1 for q in equation
1
5p (- 3) 0
5p 3
p 3/5 0.6
Solution is q -1 p 0.6
Begin Solution
Continue Solution
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28
Markers Comments
Note When the signs are the same subtract
to eliminate. When the signs are different
add to eliminate.
1. Eliminate either ps or qs by making
coefficient same.
1
2
Now get
(x4)
1

3
(x5)
2
4

4
3
-
Next Comment
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29
Markers Comments
Note When the signs are the same subtract
to eliminate. When the signs are different
add to eliminate.
1. Eliminate either ps or qs by making
coefficient same.
1
2
Now get
(x4)
1

3
(x5)
2
4

4
3
-
Next Comment
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30
Simultaneous Equations Question 3
If two coffees three doughnuts cost 2.90 while
three coffees one doughnut cost 2.60 then
find the cost of two coffees five doughnuts.
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31
Simultaneous Equations Question 3
If two coffees three doughnuts cost 2.90 while
three coffees one doughnut cost 2.60 then
find the cost of two coffees five doughnuts.
Form two equations, keeping costs in pence to
avoid decimals.
Eliminate either cs or ds by making
coefficient same.
What would you like to do now?
Substitute found value into either of original
equations.
Reveal answer only
Remember to answer the question!!!
Go to full solution
Go to Comments
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32
Simultaneous Equations Question 3
If two coffees three doughnuts cost 2.90 while
three coffees one doughnut cost 2.60 then
find the cost of two coffees five doughnuts.
Two coffees five doughnuts
3.90
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33
Question 3
1. Form two equations, keeping costs in pence to
avoid decimals.
If two coffees three doughnuts cost
2.90 while three coffees one doughnut cost
2.60 then find the cost of two coffees five
doughnuts.
Let coffees cost c pence doughnuts d pence
then we have
1
2
2. Eliminate either cs or ds by making
coefficient same.
(x1)
1

3
(x3)
2
4

Begin Solution
4
3
-
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34
Question 3
3. Substitute found value into either of original
equations.
If two coffees three doughnuts cost
2.90 while three coffees one doughnut cost
2.60 then find the cost of two coffees five
doughnuts.
Let coffees cost c pence doughnuts d pence
then we have
1
2
Substitute 70 for c in equation
2
What would you like to do now?
210 d 260
d 50
Two coffees five doughnuts
Begin Solution
(2 x 70p) (5 x 50p)
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1.40 2.50
Comments
3.90
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35
Comments
Step 1 Form the two simultaneous equations first
by introducing a letter to represent the cost of
a coffee ( c) and a different letter to
represent the cost of a doughnut (d).
1. Form two equations, keeping costs in pence to
avoid decimals.
Let coffees cost c pence doughnuts d pence
then we have
1
2
i.e. 2c 3d 290 1d 2c 260
2. Eliminate either cs or ds by making
coefficient same.
(x1)
1

3
Note change to pence eases working.
(x3)
2
4

Next Comment
4
3
-
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36
Comments
Step 2 Solve by elimination. Choose whichever
variable it is easier to make have the same
coefficient in both equations.
1. Form two equations, keeping costs in pence to
avoid decimals.
Let coffees cost c pence doughnuts d pence
then we have
1
2
2. Eliminate either cs or ds by making
coefficient same.
(x1)
1

3
(x3)
2
4

Next Comment
4
3
-
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37
Comments
Step 3 Once you have a value for one variable you
can substitute this value into any of
the equations to find the value of the other
variable. It is usually best to choose
an equation that you were given in question.
3. Substitute found value into either of original
equations.
Let coffees cost c pence doughnuts d pence
then we have
Substitute 70 for c in equation
Step 4 Remember to answer the question!!!
210 d 260
d 50
Two coffees five doughnuts
(2 x 70p) (5 x 50p)
Next Comment
1.40 2.50
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3.90
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38
Simultaneous Equations Question 3B
A company plan to introduce a new blend of
tropical fruit drink made from bananas kiwis.
They produce three blends of the drink for
market research purposes. Blend 1 uses 70
banana syrup, 30 kiwi syrup and the cost per
litre is 74p. Blend 2 has 55 banana syrup, 45
kiwi syrup and costs 71p per litre to produce.
A third blend is made using 75 banana syrup
and 25 kiwi. How does its cost compare to the
other two blends?
Get hint
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Reveal answer
EXIT
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39
Simultaneous Equations Question 3B
A company plan to introduce a new blend of
tropical fruit drink made from bananas kiwis.
They produce three blends of the drink for
market research purposes. Blend 1 uses 70
banana syrup, 30 kiwi syrup and the cost per
litre is 74p. Blend 2 has 55 banana syrup, 45
kiwi syrup and costs 71p per litre to produce.
A third blend is made using 75 banana syrup
and 25 kiwi. How does its cost compare to the
other two blends?
Eliminate either cs or ds by making
coefficient same.
Form two equations, eliminating decimals
wherever possible.
Substitute found value into either of original
equations.
Remember to answer the question!!!
What would you like to do now?
Go to Comments
Reveal answer
Go to full solution
Go to Sim Eq Menu
EXIT
40
Simultaneous Equations Question 3B
A company plan to introduce a new blend of
tropical fruit drink made from bananas kiwis.
They produce three blends of the drink for
market research purposes. Blend 1 uses 70
banana syrup, 30 kiwi syrup and the cost per
litre is 74p. Blend 2 has 55 banana syrup, 45
kiwi syrup and costs 71p per litre to produce.
A third blend is made using 75 banana syrup
and 25 kiwi. How does its cost compare to the
other two blends?
So this blend is more expensive than the other
two.
Go to Comments
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41
Question 3B
1. Form two equations, keeping costs in pence to
avoid decimals.
Blend 1 uses 70 banana syrup, 30 kiwi syrup
and the cost per litre is 74p. Blend 2 has 55
banana syrup, 45 kiwi syrup and costs 71p per
litre to produce. A third blend is made using
75 banana syrup and 25 kiwi. How does its cost
compare to the other two blends?
Let one litre of banana syrup cost B pence one
litre of kiwi syrup cost K pence.
1
2
2. Get rid of decimals
(x10)
1

3
(x100)
2
4

Begin Solution
Continue Solution
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42
Question 3B
3. Eliminate either Bs or Ks by making
coefficient same.
Blend 1 uses 70 banana syrup, 30 kiwi syrup
and the cost per litre is 74p. Blend 2 has 55
banana syrup, 45 kiwi syrup and costs 71p per
litre to produce. A third blend is made using
75 banana syrup and 25 kiwi. How does its cost
compare to the other two blends?
(x15)
3
4
(x15)
3
5

4
5
4
-
Begin Solution
Continue Solution
Comments
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43
Question 3B
4. Substitute found value into an equation
without decimals.
Blend 1 uses 70 banana syrup, 30 kiwi syrup
and the cost per litre is 74p. Blend 2 has 55
banana syrup, 45 kiwi syrup and costs 71p per
litre to produce. A third blend is made using
75 banana syrup and 25 kiwi. How does its cost
compare to the other two blends?
(x15)
3
4
Substitute 80 for B in equation
3
560 3K 740
3K 180
K 60
5. Use these values to answer question.
What would you like to do now?
75B25K
(0.75 x 80p)(0.25 x 60p)
Begin Solution
60p 15p
Continue Solution
75p
Comments
So this blend is more expensive than the other
two.
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44
Markers Comments
Step 1 Form the two simultaneous equations first
by introducing a letter to represent the cost
per litre of banana syrup ( B) and a different
letter to represent the cost per litre of kiwis
fruit (K).
1. Form two equations, keeping costs in pence to
avoid decimals.
Let one litre of banana syrup cost B pence one
litre of kiwi syrup cost K pence.
1
2
i.e. 0.70 B 0.30K 74 0.55B 0.45K
71
2. Get rid of decimals
(x10)
1

3
Multiply all terms by 100 to remove decimals.
Note change to pence eases working.
(x100)
2
4

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45
Markers Comments
Step 2 Solve by elimination. Choose whichever
variable it is easier to make have the same
coefficient in both equations.
3. Eliminate either Bs or Ks by making
coefficient same.
(x15)
3
4
(x15)
3
5

4
-
5
4
Next Comment
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46
Markers Comments
Step 3 Once you have a value for one variable you
can substitute this value into any of
the equations to find the value of the other
variable. It is usually best to choose
an equation that you were given in question.
4. Substitute found value into an equation
without decimals.
(x15)
3
4
Substitute 80 for B in equation
3
560 3K 740
3K 180
Step 4 Remember to answer the question!!!
K 60
5. Use these values to answer question.
75B25K
(0.75 x 80p)(0.25 x 60p)
60p 15p
Next Comment
75p
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So this blend is more expensive than the other
two.
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47
Simultaneous Equations Question 4

• A Fibonacci Sequence is formed as follows.
• Start with two terms add first two to obtain 3rd
• add 2nd 3rd to obtain 4th
• add 3rd 4th to obtain 5th etc
• eg starting with 3 7, next four terms are 10,
  17, 27 44.
• If the first two terms of such a sequence are P
  and Q then find expressions for the next 4 terms
  in their simplest form.
• If the 5th 6th terms are 8 and 11 respectively
  then write down two equations in P and Q and
  hence find the values of P and Q.

Get hint
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Reveal answer
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48
Write down expressions using previous two to
form next.
Simultaneous Equations Question 4
Solve and substitute found value into either of
original equations.

• A Fibonacci Sequence is formed as follows.
• Start with two terms add first two to obtain 3rd
• add 2nd 3rd to obtain 4th
• add 3rd 4th to obtain 5th etc
• eg starting with 3 7, next four terms are 10,
  17, 27 44.
• If the first two terms of such a sequence are P
  and Q then find expressions for the next 4 terms
  in their simplest form.
• If the 5th 6th terms are 8 and 11 respectively
  then write down two equations in P and Q and
  hence find the values of P and Q.

Establish two equations. Eliminate either Ps or
Qs by making coefficient same.
To find values of P Q Match term from (a) with
values given in question.
What would you like to do now?
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Reveal answer
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49
Simultaneous Equations Question 4

• A Fibonacci Sequence is formed as follows.
• Start with two terms add first two to obtain 3rd
• add 2nd 3rd to obtain 4th
• add 3rd 4th to obtain 5th etc
• eg starting with 3 7, next four terms are 10,
  17, 27 44.
• If the first two terms of such a sequence are P
  and Q then find expressions for the next 4 terms
  in their simplest form.
• If the 5th 6th terms are 8 and 11 respectively
  then write down two equations in P and Q and
  hence find the values of P and Q.

P Q,
P 2Q
2P 3Q
3P 5Q
P 7
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50
Question 4
1. Write down expressions using previous two to
form next.

• If the first two terms of such a
• sequence are P and Q then find
• expressions for the next 4 terms
• in their simplest form.

(a) First term P second term Q
3rd term P Q
4th term Q (P Q) P 2Q
5th term (P Q) (P 2Q)
2P 3Q
6th term (P 2Q) (2P 3Q)
3P 5Q
Begin Solution
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Question 4
1. Match term from (a) with values given in
question.
(b) If the 5th 6th terms are 8 and 11
respectively then write down two equations in P
and Q and hence find the values of P and Q.
(x3)
1
(x2)
2
2. Eliminate either Ps or Qs by making
coefficient same.
(x3)
1
3

2
4

(x2)
4
3
-
Begin Solution
Try another like this
Comments
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Question 4
3. Substitute found value into either of original
equations.
(b) If the 5th 6th terms are 8 and 11
respectively then write down two equations in P
and Q and hence find the values of P and Q.
(x3)
1
(x2)
2
Substitute -2 for Q in equation
2
2P (-6) 8
2P 14
P 7
First two terms are 7 and 2 respectively.
Begin Solution
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Comments
For problems in context it is often useful to do
a simple numerical example before attempting the
algebraic problem. Fibonacci Sequence 3,
7, 10, 17, 27, 44, P Q P Q P
2Q 4, 6, 10, 16, 26, 42, Then introduce
the variables P, Q, P Q, P 2Q, 2P 3Q,
1. Write down expressions using previous two to
form next.
(a) First term P second term Q
3rd term P Q
4th term Q (P Q) P 2Q
5th term (P Q) (P 2Q)
2P 3Q
6th term (P 2Q) (2P 3Q)
3P 5Q
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54
Simultaneous Equations Question 4B
In a Number Pyramid the value in each
block is the sum of the values in the two blocks
underneath ..
The two number pyramids below have the middle two
rows missing. Find the values of v and w.
(B)
(A)
-18
11
3v 2w w 2v v w
v w v 3w 6w v - w
Get hint
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Reveal answer
EXIT
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55
Simultaneous Equations Question 4B
In a Number Pyramid the value in each
block is the sum of the values in the two blocks
underneath ..
Form 2 equations and eliminate either vs or ws
by making coefficient same.
Substitute found value into either of original
equations.
Work your way to an expression for the top row
by filling in the middle rows.
The two number pyramids below have the middle two
rows missing. Find the values of v and w.
(B)
(A)
-18
11
3v 2w w 2v v w
v w v 3w 6w v - w
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Reveal answer
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56
Simultaneous Equations Question 4B
In a Number Pyramid the value in each
block is the sum of the values in the two blocks
underneath ..
The two number pyramids below have the middle two
rows missing. Find the values of v and w.
(B)
(A)
-18
11
3v 2w w 2v v w
v w v 3w 6w v - w
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V 4
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57
Question 4B
1. Work your way to an expression for the top row
by filling in the middle rows.
(A)
-18
Pyramid (A)
3v 2w w 2v v w
2nd row 3v 2w, -2v 3w, -v 2w
(B)
3rd row v 5w, -3v 5w
11
- 18
Top row -2v 10w
v w v 3w 6w v - w
Pyramid (B)
2nd row 2v - 2w, v 3w, v 5w
Begin Solution
3rd row 3v w, 2v 8w
Continue Solution
Top row 5v 9w
11
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Question 4B
2. Form 2 equations and eliminate either vs or
ws by making coefficient same.
(A)
-18
(x5)
1
3v 2w w 2v v w
(x2)
2
Now get
(B)
11
(x5)
1
3

v w v 3w 6w v - w
2
4

(x2)
4
3

Begin Solution
Continue Solution
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59
Question 4B
3. Substitute found value into either of original
equations.
(A)
-18
(x5)
1
(x2)
2
3v 2w w 2v v w
2
Substitute -1 for W in equation
(B)
11
5V (-9) 11
5V 20
v w v 3w 6w v - w
V 4
Begin Solution
Continue Solution
Comments
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60
Comments
Use diagrams given to organise working
1. Work your way to an expression for the top row
by filling in the middle rows.
Pyramid (A)
2nd row 3v 2w, -2v 3w, -v 2w
3rd row v 5w, -3v 5w
- 18
Top row -2v 10w
Pyramid (B)
2nd row 2v - 2w, v 3w, v 5w
3rd row 3v w, 2v 8w
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Top row 5v 9w
11
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61
Markers Comments
Hence equations 10w - 2v -18 9w 5v 11
1. Work your way to an expression for the top row
by filling in the middle rows.
Pyramid (A)
Solve by the method of elimination.
2nd row 3v 2w, -2v 3w, -v 2w
3rd row v 5w, -3v 5w
- 18
Top row -2v 10w
End of simultaneous Equations
Pyramid (B)
2nd row 2v - 2w, v 3w, v 5w
3rd row 3v w, 2v 8w
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Sim Eqs Menu
Top row 5v 9w
11
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