Cambridge International Mathematics

Formulae and simultaneous equations (Chapter 7) 161

EXERCISE 7E.2

1 Solve simultaneously, using substitution:

a y=3+x

5 x¡ 2 y=0

b y=x¡ 2

x+3y=6

c y=5¡x

4 x+y=5

d y=2x¡ 1

3 x¡y=6

e y=3x+4

2 x+3y=12

f y=5¡ 2 x

5 x¡ 2 y=8

2 Use the substitution method to solve simultaneously:

a x=y+2

3 x¡ 2 y=9

b x=¡1+5y

x=3¡ 5 y

c x=6¡ 3 y

3 x¡ 3 y=2

d x=1¡ 2 y

2 x+3y=4

e x=¡ 4 ¡ 2 y

2 y¡ 3 x=8

f x=¡y¡ 8

2 x¡ 4 y=5

3aTry to solve by substitution: y=2x+5 and y=2x+7.

b What is the simultaneous solution for the equations ina? Explain your answer.

4aTry to solve by substitution: y=4x+3 and 2 y=8x+6.

b How many simultaneous solutions do the equations inahave? Explain your answer.

SOLUTION BY ELIMINATION

In many problems which require the simultaneous solution of linear equations, each equation will be of

the form ax+by=c. Solution by substitution is often tedious in such situations and the method of

eliminationof one of the variables is preferred.

One method is to make the coefficients ofx(ory) thesame sizebutopposite in signand thenaddthe

equations. This has the effect ofeliminatingone of the variables.

Example 16 Self Tutor

Solve simultaneously, by elimination: 4 x+3y=2 ::::::(1)

x¡ 3 y=8 ::::::(2)

Notice that coefficients ofyare the same size but opposite in sign.

Weaddthe LHSs and the RHSs to get an equation which containsxonly.

4 x+3y=2

+ x¡ 3 y=8

5 x =10 fadding the equationsg

) x=2 fdividing both sides by 5 g

Substituting x=2 into (1) gives 4(2) + 3y=2

) 8+3y=2

) 3 y=¡ 6

) y=¡ 2

The solution is x=2 and y=¡ 2 :

Check: in (2): (2)¡3(¡2)=2+6=8 X

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Y:\HAESE\IGCSE01\IG01_07\161IGCSE01_07.CDR Monday, 15 September 2008 4:44:17 PM PETER