Basic Engineering Mathematics, Fifth Edition

140 Basic Engineering Mathematics

isv=mt+c. But, from above, gradient,m= 2 .5and

v-axis intercept,c= 5 .5.

Hence,the equation of the graph isv= 2. 5 t+ 5. 5

Problem 11. Determine the gradient of the

straight line graph passing through the co-ordinates

(a)(− 2 , 5 )and( 3 , 4 ),and(b)(− 2 ,− 3 )and(− 1 , 3 )

From Figure 17.16, a straight line graph passing

throughco-ordinates(x 1 ,y 1 )and(x 2 ,y 2 )has a gradient

given by

m=

y 2 −y 1

x 2 −x 1

(x 1 , y 1 )

(x 2 , y 2 )

(x 2 x 1 )

(y 2  y 1 )

0

y

y 2

y 1

x 1 x 2 x

Figure 17.16

(a) A straight line passes through(− 2 , 5 )and( 3 , 4 ),

hencex 1 =−2,y 1 =5,x 2 =3andy 2 =4, hence,

gradient,m=

y 2 −y 1

x 2 −x 1

=

4 − 5

3 −(− 2 )

=−

1

5

(b) A straight line passes through (− 2 ,− 3 ) and

(− 1 , 3 ), hencex 1 =− 2 ,y 1 =−3,x 2 =−1and

y 2 =3, hence,gradient,

m=

y 2 −y 1

x 2 −x 1

=

3 −(− 3 )

− 1 −(− 2 )

=

3 + 3

− 1 + 2

=

6

1

= 6

Now try the following Practice Exercise

PracticeExercise 68 Gradients, intercepts

and equations of graphs (answerson page

347)

1. The equationofa lineis 4y= 2 x+5. A table
   of corresponding values is produced and is
   shown below. Complete the table and plot a
   graph ofyagainstx. Find the gradient of the
   graph.

x − 4 − 3 − 2 − 1 0 1 2 3 4

y − 0. 25 1.25 3.25

2. Determine the gradient and intercept on the
   y-axis for each of the following equations.
   (a) y= 4 x−2(b)y=−x
   (c) y=− 3 x−4(d)y= 4


3. Find the gradient and intercept on they-axis
   for each of the following equations.
   (a) 2y− 1 = 4 x (b) 6x− 2 y= 5
   (c) 3( 2 y− 1 )=

x

4

Determine the gradient andy-axis intercept

for each of the equations in problems 4 and

5 and sketch the graphs.

4. (a)y= 6 x−3(b)y=− 2 x+ 4
   (c)y= 3 x (d)y= 7


5. (a) 2y+ 1 = 4 x (b) 2x+ 3 y+ 5 = 0

(c) 3( 2 y− 4 )=

x

3

(d) 5x−

y

2

−

7

3

= 0

6. Determine the gradient of the straight line
   graphs passing through the co-ordinates:
   (a) (2, 7) and(− 3 , 4 )
   (b)(− 4 ,− 1 )and(− 5 , 3 )

(c)

(

1

4

,−

3

4

)

and

(

−

1

2

,

5

8

)

7. State which of the following equations will
   produce graphs which are parallel to one
   another.
   (a) y− 4 = 2 x (b) 4x=−(y+ 1 )
   (c) x=

1

2

(y+ 5 ) (d) 1+

1

2

y=

3

2

x

(e) 2x=

1

2

( 7 −y)

8. Draw on the same axes the graphs of
   y= 3 x−5and3y+ 2 x=7. Find the co-
   ordinates of the point of intersection. Check
   the result obtained by solving the two simul-
   taneous equations algebraically.


9. Plot the graphsy= 2 x+3and2y= 15 − 2 x
   on the same axes and determine their pointof
   intersection.


10. Draw on the same axes the graphs of
    y= 3 x−1andy+ 2 x=4. Find the co-
    ordinates of the point of intersection.