Basic Engineering Mathematics, Fifth Edition

156 Basic Engineering Mathematics

Rearranging each equation intoy=mx+cform gives

y=− 1. 20 x+ 1. 80 (1)

y=

x

5. 0

−

8. 5

5. 0

i.e. y= 0. 20 x− 1. 70 (2)

Three co-ordinates are calculated for each equation as

shown below.

x 0 1 2

y=− 1. 20 x+ 1. 80 1.80 0.60 − 0. 60

x 0 1 2

y= 0. 20 x− 1. 70 − 1. 70 − 1. 50 − 1. 30

The two sets of co-ordinates are plotted as shown in

Figure 19.2. The point of intersection is( 2. 50 ,− 1. 20 ).

Hence, the solution of the simultaneous equations is

x= 2. 50 ,y=− 1. 20

(It is sometimes useful to initially sketch thetwostraight

lines to determine the region where the point of inter-

section is. Then, for greater accuracy, a graph having a

smaller range of values can be drawn to ‘magnify’ the

point of intersection.)

21

21

22

23

2 1.20

23 22

3

1

0 123

2.50

4

y

x

y 5 0.20x 2 1.70

y 52 1.20x 1 1.80

Figure 19.2

Now try the following Practice Exercise

PracticeExercise 72 Graphical solution of

simultaneous equations (Answers on

page 347)

In problems 1 to 6, solve the simultaneous equa-

tions graphically.

1.y= 3 x−22.x+y= 2

y=−x+ 63 y− 2 x= 1

3. y= 5 −x 4. 3x+ 4 y= 5
   x−y= 22 x− 5 y+ 12 = 0


4. 
   
   
   
   1. 4 x− 7. 06 = 3. 2 y 6. 3x− 2 y= 0
   
   
   2. 1 x− 6. 7 y= 12. 87 4 x+y+ 11 = 0
   
   
   
   


5. The friction force F newtons and load L
   newtons are connected by a law of the form
   F=aL+b,wherea and b are constants.
   WhenF=4N,L=6N and whenF= 2 .4N,
   L=2N. Determine graphically the values ofa
   andb.

19.2 Graphical solution of quadratic equations

A general quadratic equation is of the form

y=ax^2 +bx+c,wherea,bandcare constants and

ais not equal to zero.

A graph of a quadratic equation always produces a shape

called aparabola.

The gradients of the curves between 0 and A and

betweenBandCin Figure 19.3 are positive, whilst

the gradient betweenAandBis negative. Points such

asAandBare calledturning points.AtAthe gradi-

ent is zero and, asxincreases, the gradient of the curve

changesfrompositivejust beforeAtonegativejustafter.

Such a point is called amaximum value.AtBthe gra-

dient is also zero and, asxincreases, the gradient of the

curve changes from negative just beforeBto positive

just after. Such a point is called aminimum value.

y

x

A C

B

0

Figure 19.3

Following are three examples of solutions using

quadratic graphs.

(a) y=ax^2

Graphs of y=x^2 ,y= 3 x^2 and y=

1

2

x^2 are

shown in Figure 19.4. All have minimum values at

the origin (0, 0).