Basic Engineering Mathematics, Fifth Edition

Straight line graphs 137

given by

FE

ED

=

− 5 −(− 17 )

0 − 3

=

12

− 3

=− 4

Hence, the gradient of both y=− 4 x+4and
y=− 4 x−5is− 4 , which, again, could have been
deduced ‘on sight’.
They-axis intercept means the value ofywhere the
straight line cuts they-axis. From Figure 17.10,

y= 3 xcuts they-axis aty= 0

y= 3 x+7 cuts they-axis aty=+ 7

y=− 4 x+4 cuts they-axis aty=+ 4

y=− 4 x−5 cuts they-axis aty=− 5

Some general conclusionscan be drawn from the graphs
shown in Figures 17.9 and 17.10. When an equation is
of the formy=mx+c,wheremandcare constants,
then

(a) agraph ofyagainstxproduces a straight line,

(b) mrepresents the slope or gradient of the line, and

(c) crepresents they-axis intercept.

Thus, given an equation such asy= 3 x+7, it may be
deduced ‘on sight’ that its gradient is+3 and itsy-axis
intercept is+7, as shown in Figure 17.10. Similarly, if
y=− 4 x−5, the gradient is−4andthey-axis intercept
is−5, as shown in Figure 17.10.
When plottinga graphof theformy=mx+c, onlytwo
co-ordinates need bedetermined.When theco-ordinates
are plotted a straight line is drawn between the two
points. Normally, three co-ordinates are determined, the
third one acting as a check.

Problem 4. Plot the graph 3x+y+ 1 =0and

2 y− 5 =xon the same axes and find their point of

intersection

Rearranging 3x+y+ 1 =0givesy=− 3 x− 1

Rearranging 2 y− 5 =x gives 2y=x+5and

y=

1

2

x+ 2

1

2

Since both equations are of the formy=mx+c, both
are straight lines. Knowing an equation is a straight
line means that only two co-ordinates need to be plot-
ted and a straight line drawn through them. A third
co-ordinate is usually determined to act as a check.

A table of values is produced for each equation as shown

below.

x 1 0 − 1

− 3 x− 1 − 4 − 1 2

x 2 0 − 3

1

2 x+^2

1

2 3

1

2 2

1

2 1

The graphs are plotted as shown in Figure 17.11.

Thetwo straight lines are seen to intersect at(− 1 , 2 ).

4

24 23 22 210

3

2

1

21

22

23

24

y 523 x 21

y

1 2 3 4 x

y 5 12 x (^152)
Figure 17.11
Problem 5. If graphs ofyagainstxwere to be
plotted for each of the following, state (i) the
gradient and (ii) they-axis intercept.
(a)y= 9 x+2(b)y=− 4 x+ 7
(c)y= 3 x (d)y=− 5 x− 3
(e)y=6(f)y=x
If y=mx+c then m=gradient and c=y-axis
intercept.
(a) Ify= 9 x+2, then (i)gradient= 9
(ii)y-axis intercept= 2
(b) Ify=− 4 x+7, then (i)gradient=− 4
(ii)y-axis intercept= 7
(c) Ify= 3 xi.e.y= 3 x+0,
then (i)gradient= 3
(ii)y-axis intercept= 0
i.e. the straight line passes through the origin.