1
00:00:00,640 --> 00:00:03,120
NARRATOR: Exercise 6. Part a.

2
00:00:03,120 --> 00:00:06,480
Find the equation of the line
parallel to the x-axis

3
00:00:06,480 --> 00:00:08,960
which passes through the point
where the lines

4
00:00:08,960 --> 00:00:15,720
4x plus 3y minus 6 equals 0
and x minus 2y minus 7 equals 0 meet.

5
00:00:15,720 --> 00:00:17,200
Part b.

6
00:00:17,200 --> 00:00:19,400
Find the gradient of the line

7
00:00:19,400 --> 00:00:21,880
which passes through the point
(2, negative 3)

8
00:00:21,880 --> 00:00:24,120
and the point of intersection
of the lines

9
00:00:24,120 --> 00:00:29,400
3x plus 2y equals 2
and 4x plus 3y equals 7.

10
00:00:30,600 --> 00:00:32,080
Part a.

11
00:00:32,080 --> 00:00:34,640
Find the equation of the line
parallel to the x-axis

12
00:00:34,640 --> 00:00:38,360
which passes through the point of
intersection of the two given lines.

13
00:00:38,360 --> 00:00:42,720
If the line is parallel to the
x-axis, then it has a gradient of 0.

14
00:00:42,720 --> 00:00:46,280
It passes through the point of
intersection between the lines

15
00:00:46,280 --> 00:00:52,000
4x plus 3y minus 6 equals 0
and x minus 2y minus 7 equals 0.

16
00:00:52,000 --> 00:00:56,160
These equations can be solved
simultaneously to find that point.

17
00:00:58,160 --> 00:01:01,440
Calling the equations
'one' and 'two' respectively,

18
00:01:01,440 --> 00:01:03,520
and then multiplying
equation two by 4

19
00:01:03,520 --> 00:01:06,400
in order to create the same x term
in both equations

20
00:01:06,400 --> 00:01:11,800
allows us then to subtract to get
the equation 11y plus 22 equals 0,

21
00:01:11,800 --> 00:01:15,800
so 11y equals negative 22
and y equals negative 2.

22
00:01:15,800 --> 00:01:20,120
Substituting y equals negative 2
into the second equation

23
00:01:20,120 --> 00:01:21,920
gives x equals 3.

24
00:01:21,920 --> 00:01:26,720
And so the line passes through
the point (3, negative 2).

25
00:01:26,720 --> 00:01:28,960
The line has a gradient of 0

26
00:01:28,960 --> 00:01:31,480
and passes through the point
(3, negative 2).

27
00:01:31,480 --> 00:01:35,000
A line with a gradient of 0
is of the form y equals c.

28
00:01:35,000 --> 00:01:41,000
And in this case the required line
has equation y equals negative 2.

29
00:01:41,000 --> 00:01:42,480
Part b.

30
00:01:42,480 --> 00:01:44,280
Find the gradient of the line

31
00:01:44,280 --> 00:01:46,560
which passes through the point
(2, negative 3)

32
00:01:46,560 --> 00:01:49,440
and the point of intersection
of the given lines.

33
00:01:49,440 --> 00:01:52,480
We must first find the point of
intersection of the lines

34
00:01:52,480 --> 00:01:57,320
3x plus 2y equals 2
and 4x plus 3y equals 7.

35
00:01:57,320 --> 00:02:00,720
Calling these equations
'one' and 'two' respectively

36
00:02:00,720 --> 00:02:05,160
and then multiplying equation one
by 4 and equation two by three

37
00:02:05,160 --> 00:02:08,440
in order to create the same x term
in both equations.

38
00:02:08,440 --> 00:02:12,520
Subtracting these new equations
gives y equals 13.

39
00:02:12,520 --> 00:02:15,840
And substituting y equals 13
into the first equation

40
00:02:15,840 --> 00:02:17,720
gives x equals negative 8.

41
00:02:19,120 --> 00:02:22,920
So the lines intersect at the point
(negative 8, 13).

42
00:02:22,920 --> 00:02:25,760
It's now been established
that we wish to find

43
00:02:25,760 --> 00:02:27,760
the gradient of the
line joining the points

44
00:02:27,760 --> 00:02:30,520
(2, negative 3) and (negative 8, 13).

45
00:02:30,520 --> 00:02:32,760
Gradient is equal to rise over run,

46
00:02:32,760 --> 00:02:34,960
that is the difference
in the y-coordinates

47
00:02:34,960 --> 00:02:37,160
divided by the difference
in the x-coordinates.

48
00:02:37,160 --> 00:02:39,120
And so in this case, we have

49
00:02:39,120 --> 00:02:42,880
13 minus negative 3 divided by
negative eight minus 2,

50
00:02:42,880 --> 00:02:45,440
giving 16 over negative 10,

51
00:02:45,440 --> 00:02:47,120
which is equivalent

52
00:02:47,120 --> 00:02:48,760
to negative eight-fifths

53
00:02:48,760 --> 00:02:50,680
or negative 1.6.