Basic Engineering Mathematics, Fifth Edition

138 Basic Engineering Mathematics

(d) Ify=− 5 x−3, then (i)gradient=− 5

(ii)y-axis intercept=− 3

(e) If y=6i.e.y= 0 x+6,

then (i)gradient= 0

(ii)y-axis intercept= 6

i.e.y=6 is a straight horizontal line.

(f) Ify=xi.e.y= 1 x+0,

then (i)gradient= 1

(ii)y-axis intercept= 0

Sincey=x,asxincreases,yincreases by the same

amount; i.e.,yis directly proportional tox.

Problem 6. Without drawing graphs, determine

the gradient andy-axis intercept for each of the

following equations.

(a)y+ 4 = 3 x (b) 2y+ 8 x=6(c)3x= 4 y+ 7

If y=mx+c then m=gradient and c=y-axis

intercept.

(a) Transposingy+ 4 = 3 xgives y= 3 x− 4

Hence,gradient= 3 andy-axis intercept=− 4.

(b) Transposing 2y+ 8 x=6gives 2y=− 8 x+ 6

Dividing both sides by 2 gives y=− 4 x+ 3

Hence,gradient=− 4 andy-axis intercept= 3.

(c) Transposing 3x= 4 y+7gives3x− 7 = 4 y

or 4 y= 3 x− 7

Dividing both sides by 4 gives y=

3

4

x−

7

4

or y= 0. 75 x− 1. 75

Hence,gradient =0.75and y-axis intercept

=− 1. 75

Problem 7. Without plotting graphs, determine

the gradient andy-axis intercept values of the

following equations.

(a)y= 7 x−3(b)3y=− 6 x+ 2

(c)y− 2 = 4 x+9(d)

y

3

=

x

3

−

1

5

(e) 2x+ 9 y+ 1 = 0

(a) y= 7 x−3 is of the formy=mx+c

Hence,gradient,m= 7 andy-axis intercept,

c=− 3.

(b) Rearranging 3y=− 6 x+2gives y=−

6 x

3

+

2

3

,

i.e. y=− 2 x+

2

which is of the formy=mx+c^3

Hence,gradientm=− 2 andy-axis intercept,

c=

2

3

(c) Rearrangingy− 2 = 4 x+9givesy= 4 x+11.

Hence,gradient= 4 andy-axis intercept= 11.

(d) Rearranging

y

3

=

x

2

−

1

5

gives y= 3

(

x

2

−

1

5

)

=

3

2

x−

3

5

Hence,gradient=

3

2

and

y-axis intercept=−

3

5

(e) Rearranging 2x+ 9 y+ 1 =0gives9y=− 2 x−1,

i.e. y=−

2

9

x−

1

9

Hence,gradient=−

2

9

and

y-axis intercept=−

1

9

Problem 8. Determine for the straight line shown

in Figure 17.12 (a) the gradient and (b) the equation

of the graph

23

20

24 23 220

15

10

8

5

25

210

215

220

y

21 1342 x

Figure 17.12

(a) A right-angled triangleABCis constructed on the

graph as shown in Figure 17.13.

Gradient=

AC

CB

=

23 − 8

4 − 1

=

15

3

= 5