Cambridge International Mathematics

Formulae and simultaneous equations (Chapter 7) 159

In this course we will considerlinear simultaneous equationscontaining two unknowns, usuallyxandy.

There are infinitely many values ofxandywhich satisfy the first equation. Likewise, there are infinitely

many values ofxandywhich satisfy the second equation. In general, however, only one combination of

values forxandysatisfiesbothequations at the same time.

For example, consider the simultaneous equations

½

x+y=9

2 x+3y=21

.

If x=6 and y=3 then:

² x+y= (6) + (3) = 9 X The first equation is satisfied

² 2 x+3y= 2(6) + 3(3) = 12 + 9 = 21 X The second equation is satisfied.

So, x=6andy=3is thesolutionto the simultaneous equations

½

x+y=9

2 x+3y=21

.

We thus consideralgebraic methodsfor finding the simultaneous solution.

EQUATING VALUES OFy

If both equations are given withyas thesubject, we can find the simultaneous solution by equating the

right hand sides of the equations.

Example 13 Self Tutor

Find the simultaneous solution to the equations: y=2x¡ 1 , y=x+3

If y=2x¡ 1 and y=x+3, then

2 x¡1=x+3 fequatingysg

) 2 x¡ 1 ¡x=x+3¡x fsubtractingxfrom both sidesg

) x¡1=3

) x=4 fadding 1 to both sidesg

and so y=4+3 fusing y=x+3g

) y=7

So, the simultaneous solution is x=4 and y=7.

Check: In y=2x¡ 1 , y=2£ 4 ¡1=8¡1=7 X

In y=x+3, y=4+3=7 X

EXERCISE 7E.1

1 Find the simultaneous solution to the following pairs of equations:

a y=x¡ 2

y=3x+6

b y=x+2

y=2x¡ 3

c y=6x¡ 6

y=x+4

d y=2x+1

y=x¡ 3

e y=5x+2

y=3x¡ 2

f y=3x¡ 7

y=3x¡ 2

Always check

your solution in

both equations.

The solutions to linear simultaneous equations can be found bytrial and erroras in theDiscovery, but this

can be quite tedious. They may also be found by drawinggraphs, but this can be slow and also inaccurate

if the solutions are not integers.

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Y:\HAESE\IGCSE01\IG01_07\159IGCSE01_07.CDR Tuesday, 18 November 2008 11:57:42 AM PETER