138 Basic Engineering Mathematics
(d) If y = −5x − 3, then (i) gradient = −5
(ii) y-axis intercept = −3
(e) If y = 6 i.e. y = 0x + 6,
then
(i) gradient = 0
(ii) y-axis intercept = 6
i.e. y = 6 is a straight horizontal line.
(f ) If y = x i.e. y = 1x + 0,
then
(i) gradient = 1
(ii) y-axis intercept = 0
Since y = x, as x increases, y increases by the same
amount; i.e., y is directly proportional to x.
Problem 6. Without drawing graphs, determine
the gradient and y-axis intercept for each of the
following equations.
(a) y + 4 = 3x (b) 2y + 8x = 6 (c) 3x = 4y + 7
If y = mx + c then m = gradient and c = y-axis
intercept.
(a) Transposing y + 4 = 3x gives y = 3x − 4
Hence, gradient = 3 and y-axis intercept = −4.
(b) Transposing 2y + 8x = 6 gives 2y = −8x + 6
Dividing both sides by 2 gives y = −4x + 3
Hence, gradient = −4 and y-axis intercept = 3.
(c) Transposing 3x = 4y + 7 gives 3x − 7 = 4y
or
4y = 3x − 7
Dividing both sides by 4 gives y=
3
4
x −
7
4
or
y= 0.75x − 1.75
Hence, gradient = 0.75 and y-axis intercept
= −1.75
Problem 7. Without plotting graphs, determine
the gradient and y-axis intercept values of the
following equations.
(a) y = 7x − 3
( b )3 y = −6x + 2
(c) y − 2 = 4x + 9
( d )
y
3
=
x
3
−
1
5
(e) 2x + 9y + 1 = 0
(a) y = 7x − 3 is of the form y = mx + c
Hence, gradient, m = 7 and y-axis intercept,
c = −3.
(b) Rearranging 3y = −6x + 2 gives y = −
6x
3
+
2
3
,
i.e.
y = −2x +
2
3
which is of the form y = mx + c
Hence, gradient m = −2 and y-axis intercept,
c =
2
3
(c) Rearranging y − 2 = 4x + 9 gives y = 4x + 11.
Hence, gradient = 4 and y-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 + 9y + 1 =0 gives 9y = − 2x − 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 22
0
15
10
8
5
25
210
215
220
y
x
1
3
4
2
21
Figure 17.12
(a) A right-angled triangle ABC is constructed on the
graph as shown in Figure 17.13.
Gradient =
AC
CB
=
23 − 8
4 − 1
=
15
3
= 5
(d) If y = −5x − 3, then (i) gradient = −5
(ii) y-axis intercept = −3
(e) If y = 6 i.e. y = 0x + 6,
then
(i) gradient = 0
(ii) y-axis intercept = 6
i.e. y = 6 is a straight horizontal line.
(f ) If y = x i.e. y = 1x + 0,
then
(i) gradient = 1
(ii) y-axis intercept = 0
Since y = x, as x increases, y increases by the same
amount; i.e., y is directly proportional to x.
Problem 6. Without drawing graphs, determine
the gradient and y-axis intercept for each of the
following equations.
(a) y + 4 = 3x (b) 2y + 8x = 6 (c) 3x = 4y + 7
If y = mx + c then m = gradient and c = y-axis
intercept.
(a) Transposing y + 4 = 3x gives y = 3x − 4
Hence, gradient = 3 and y-axis intercept = −4.
(b) Transposing 2y + 8x = 6 gives 2y = −8x + 6
Dividing both sides by 2 gives y = −4x + 3
Hence, gradient = −4 and y-axis intercept = 3.
(c) Transposing 3x = 4y + 7 gives 3x − 7 = 4y
or
4y = 3x − 7
Dividing both sides by 4 gives y=
3
4
x −
7
4
or
y= 0.75x − 1.75
Hence, gradient = 0.75 and y-axis intercept
= −1.75
Problem 7. Without plotting graphs, determine
the gradient and y-axis intercept values of the
following equations.
(a) y = 7x − 3
( b )3 y = −6x + 2
(c) y − 2 = 4x + 9
( d )
y
3
=
x
3
−
1
5
(e) 2x + 9y + 1 = 0
(a) y = 7x − 3 is of the form y = mx + c
Hence, gradient, m = 7 and y-axis intercept,
c = −3.
(b) Rearranging 3y = −6x + 2 gives y = −
6x
3
+
2
3
,
i.e.
y = −2x +
2
3
which is of the form y = mx + c
Hence, gradient m = −2 and y-axis intercept,
c =
2
3
(c) Rearranging y − 2 = 4x + 9 gives y = 4x + 11.
Hence, gradient = 4 and y-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 + 9y + 1 =0 gives 9y = − 2x − 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 22
0
15
10
8
5
25
210
215
220
y
x
1
3
4
2
21
Figure 17.12
(a) A right-angled triangle ABC is constructed on the
graph as shown in Figure 17.13.
Gradient =
AC
CB
=
23 − 8
4 − 1
=
15
3
= 5