© John Wiley & Sons Australia, Ltd Page 1

WorkSHEET 4.2 Simultaneous equations II Name: ___________________________ 1 Solve the following pair of simultaneous

equations using the substitution method:

1

2 3 17x yx y= -- =

( )( )

( )( )

( )( )

1 1

2 3 17 2

Substitute (1) into 2

2 1 3 172 2 3 17

5 153

Substitute 3 into 1

1 34

x y

x y

y yy y

yyy

xx

= -

- =

- - =

- - =- =

= -

= -

= - -

=

Solution: (4, -3)

2 Solve the following pair of simultaneous equations using the elimination method:

156364

=-=-yxyx

( )

4 63 6 15

(1) 3: 3 12 18(2) (3): 6 3

12

1Substitute into 12

14 622 6

41Solution: 4,2

x yx yx y

y

y

y

x

xx

- =- =

´ - =- = -

= -

= -

- ´- =

+ ==

æ ö-ç ÷è ø

)3()2()1(

© John Wiley & Sons Australia, Ltd Page 2

3 Solve the following pair of simultaneous equations using the elimination method:

463794-=+-=+

yxyx

( )

÷øö

çèæ -

==-=-

-=-´+

-=

÷øö

çèæ--=

-=--=+´-=+´-=+-=+

321,2:Solution

284

7154

73594

1 into35substitute

321

3553:(4) (3)

)4(162412:4 (2))3(212712:3 (1)

)2(463)1(794

xx

x

x

y

y

yyxyxyxyx

4 Find two numbers whose sum is 23 and whose difference is 27.

Let the two numbers be x and y. ( )( )

( )

23 1

27 2(1) (2) : 2 50

25Substitute 25 into 1

25 232

x y

x yxxxyy

+ =

- =

+ ==

=

+ == -

The two numbers are 25 and -2.

5 A rectangular swimming pool has a perimeter of 50 metres. The length is 9 metres more than the width. What are the dimensions of the pool?

Let x be the width of the pool. The length is x + 9. Perimeter = 50

( )

8432

18225092250

==

++=++=

xx

xxxx

Width is 8 metres. Length is 17 metres.

© John Wiley & Sons Australia, Ltd Page 3

6 A moneybox contains only 10c and 20c coins. If there are 54 coins altogether, totalling $8.60, how many of each type of coin are there?

Let x be the number of 10c coins. Let y be the number of 20c coins.

( )( )( )

( )

54 1

10 20 860 2

(2) 10: 2 86 3(3) (1): 32

Substitute 32 into 132 54

22

x y

x y

x yyy

xx

+ =

+ =

÷ + =

- =

=

+ ==

Moneybox contains 22 × 10c coins and 32 × 20c coins.

7 Nicholas buys 4 pears and 3 apples for $2.75. Nadia buys 7 pears and 5 apples for $4.70. How much does each type of fruit cost?

Let p represent the cost of a pear and a represent the cost of an apple (in cents).

( )( )( )( )

( )

4 3 275 1

7 5 470 2

(1) 5: 20 15 1375 3

(2) 3: 21 15 1410 4(4) (3): 35Substitute 35 into 1

4 35 3 275140 3 275

3 13545

p a

p a

p a

p appaaaa

+ =

+ =

´ + =

´ + =

- =

=

´ + =+ =

==

A pear costs 35 cents and an apple 45 cents.

© John Wiley & Sons Australia, Ltd Page 4

8 Three adults and 5 children pay $62.30 to enter the zoo, whereas 4 adults and 3 children pay $60.70. What was the entry fee for each adult and each child?

Let a represent an adult’s entry fee and c represent a child’s entry fee (in cents).

( )( )( )( )

( )

3 5 6230 1

4 3 6070 2

(1) 3: 9 15 18 690 3

(2) 5: 20 15 30350 4(4) (3) : 11 11 660

1060Substitute 1060 into 1

3 1060 5 62303180 5 6230

5 3050610

a c

a c

a c

a caaacccc

+ =

+ =

´ + =

´ + =

- ==

=

´ + =+ =

==

Entrance fees are adults $10.60 and children $6.10.

9 Consider three integers a, b and c. If a + b = −3 b + c = 1 and c + a = −6 determine the values of a, b and c.

3 (1)1 (2)

6 (3)(1) (2) :

4 (4)(3) (4) :

2 105

Substitute 5 into (1)5 3

2Substitute 5 into (3)

5 61

a bb cc a

a c

aaabba

cc

+ = -+ =+ = --- = -+

= -= -= -

- + = -== -

- = -= -

So, a = −5, b = 2 and c = −1.