1
00:00:00,000 --> 00:00:00,380


2
00:00:00,380 --> 00:00:04,655
Let's do some equations of lines in point-slope form.

3
00:00:04,655 --> 00:00:07,540


4
00:00:07,540 --> 00:00:12,644
And this is different from y or from slope-intercept form.

5
00:00:12,644 --> 00:00:15,649


6
00:00:15,650 --> 00:00:18,199
But they really are just two different ways of writing the

7
00:00:18,199 --> 00:00:19,329
same equation.

8
00:00:19,329 --> 00:00:21,219
We'll see that with a couple of examples.

9
00:00:21,219 --> 00:00:24,449
And you might remember that slope-intercept form were

10
00:00:24,449 --> 00:00:30,030
equations of the form, y is equal to mx plus b-- we did

11
00:00:30,030 --> 00:00:33,280
this in the last video-- where m is the slope, b is the

12
00:00:33,280 --> 00:00:34,850
intercept, the y-intercept.

13
00:00:34,850 --> 00:00:36,060
That's why it's called slope-intercept.

14
00:00:36,060 --> 00:00:38,320
You have the slope and the intercept.

15
00:00:38,320 --> 00:00:46,649
In point-slope form, it takes the form y minus y1-- and I'll

16
00:00:46,649 --> 00:00:52,679
tell you what y1 is in a second-- is equal to m times x

17
00:00:52,679 --> 00:01:00,590
minus x1, where the coordinate x1, y1 is a point on the line.

18
00:01:00,590 --> 00:01:03,530
That's why this is called point-slope form.

19
00:01:03,530 --> 00:01:06,439
Now, these are just two different ways of writing the

20
00:01:06,439 --> 00:01:07,519
exact same thing.

21
00:01:07,519 --> 00:01:10,959
You can always algebraically manipulate this to get that,

22
00:01:10,959 --> 00:01:13,869
or algebraically manipulate that to get that.

23
00:01:13,870 --> 00:01:15,800
And I'll show you that with a couple of examples.

24
00:01:15,799 --> 00:01:18,959
But let's just do a few in point-slope form, just to make

25
00:01:18,959 --> 00:01:21,140
things concrete in your head.

26
00:01:21,140 --> 00:01:25,030
So here we have a line that has a slope of negative 1 over

27
00:01:25,030 --> 00:01:29,159
10, so m is equal to negative 1 over 10.

28
00:01:29,159 --> 00:01:32,459
And it goes through the point 10 comma 2.

29
00:01:32,459 --> 00:01:36,319
So we can directly go to point-slope form.

30
00:01:36,319 --> 00:01:37,429
So let's get a point.

31
00:01:37,430 --> 00:01:41,470
A point here is the point 10 comma 2.

32
00:01:41,469 --> 00:01:45,469
So we can immediately go to point-slope form. y minus,

33
00:01:45,469 --> 00:01:47,730
this is a y-value that's on the line.

34
00:01:47,730 --> 00:01:53,730
So y minus 2 is going to be equal to the slope, negative 1

35
00:01:53,730 --> 00:02:01,530
over 10 times x minus an x-value, times x minus 10,

36
00:02:01,530 --> 00:02:02,019
just like that.

37
00:02:02,019 --> 00:02:02,780
And we're done.

38
00:02:02,780 --> 00:02:05,650
We just put it in point-slope form.

39
00:02:05,650 --> 00:02:08,409
There's two things I want to point out to you.

40
00:02:08,409 --> 00:02:09,609
One, why this makes sense.

41
00:02:09,610 --> 00:02:11,150
And I also want to show you that this is

42
00:02:11,150 --> 00:02:12,300
equivalent to that.

43
00:02:12,300 --> 00:02:14,910
So the first thing is why does this make sense?

44
00:02:14,909 --> 00:02:17,919
Well, all this is saying, if you divide both sides of this

45
00:02:17,919 --> 00:02:24,709
equation by that right over there, you get y2 over x minus

46
00:02:24,710 --> 00:02:27,879
10 is equal to negative 1 over 10.

47
00:02:27,879 --> 00:02:32,539
Or you get the change in y between any point and 2, and

48
00:02:32,539 --> 00:02:36,379
the point 2, over the change in x.

49
00:02:36,379 --> 00:02:40,400
So you get the change in y over the change in x for any

50
00:02:40,400 --> 00:02:46,200
point x, y on that line, relative to the point 10 comma

51
00:02:46,199 --> 00:02:48,689
2, is going to be negative 1 over 10.

52
00:02:48,689 --> 00:02:52,370
This is just the definition of your slope.

53
00:02:52,370 --> 00:02:53,580
Hopefully I don't confuse you there.

54
00:02:53,580 --> 00:02:55,719
I'm just showing you that this is just using the definition

55
00:02:55,719 --> 00:02:59,189
of the slope to create the equation of the line.

56
00:02:59,189 --> 00:03:01,139
Now, the other thing I want to show is that this is

57
00:03:01,139 --> 00:03:03,099
completely equivalent to this.

58
00:03:03,099 --> 00:03:06,109
We can just algebraically manipulate this to get that.

59
00:03:06,110 --> 00:03:07,060
And let's do that.

60
00:03:07,060 --> 00:03:10,280
So this right here is the answer to the problem.

61
00:03:10,280 --> 00:03:12,340
But let's play around with it algebraically to

62
00:03:12,340 --> 00:03:13,780
get it in that form.

63
00:03:13,780 --> 00:03:17,169
So if we get y minus 2 is equal to-- let's distribute

64
00:03:17,169 --> 00:03:18,719
the negative 1 over 10.

65
00:03:18,719 --> 00:03:22,639
So it's negative 1 over 10 x, and then negative 1 over 10

66
00:03:22,639 --> 00:03:26,019
times negative 10 is plus 1.

67
00:03:26,020 --> 00:03:28,990
Now we can add 2 to both sides of this equation.

68
00:03:28,990 --> 00:03:34,850
And you get y is equal to negative 1 over 10 x plus 3.

69
00:03:34,849 --> 00:03:37,370
So just algebraically manipulating it, we were able

70
00:03:37,370 --> 00:03:39,830
to put it into slope-intercept form.

71
00:03:39,830 --> 00:03:43,750
So these two things are completely equivalent.

72
00:03:43,750 --> 00:03:47,680
Let's do a couple more problems. The line contains

73
00:03:47,680 --> 00:03:50,629
the points 10 comma 12 and 5, 25.

74
00:03:50,629 --> 00:03:53,039
So let's figure out the slope.

75
00:03:53,039 --> 00:03:59,060
So the slope, which is equal to change in y over change in

76
00:03:59,060 --> 00:04:03,000
x, is equal to-- well, let's just use this point first. So

77
00:04:03,000 --> 00:04:19,870
let's say it's 12 minus 25 over 10 minus 5.

78
00:04:19,870 --> 00:04:25,759
This is going to be equal to-- 12 minus 25 is negative 13,

79
00:04:25,759 --> 00:04:29,529
over 10 minus 5, which is 5.

80
00:04:29,529 --> 00:04:32,269
So the slope here is negative 13 over 5.

81
00:04:32,269 --> 00:04:34,979
Let's put it in point-slope form, the equation.

82
00:04:34,980 --> 00:04:39,040
So it's going to be y minus-- let's use this point right

83
00:04:39,040 --> 00:04:46,439
here-- y minus 25 is equal to the slope negative 13 over 5

84
00:04:46,439 --> 00:04:51,689
times x minus this point, 5.

85
00:04:51,689 --> 00:04:56,350
We just knew that the point 5 comma 25 is on the line.

86
00:04:56,350 --> 00:05:00,890
So y minus 25 is equal to the slope, which we figured out,

87
00:05:00,889 --> 00:05:02,849
times x minus 5.

88
00:05:02,850 --> 00:05:04,060
And we're done.

89
00:05:04,060 --> 00:05:04,649
That's all.

90
00:05:04,649 --> 00:05:07,579
If you want, out of interest, you could do the algebra to

91
00:05:07,579 --> 00:05:10,469
put this into the slope-intercept form, to the

92
00:05:10,470 --> 00:05:13,930
mx plus b form, and see that they are completely

93
00:05:13,930 --> 00:05:16,459
equivalent.

94
00:05:16,459 --> 00:05:18,769
Let's do another one.

95
00:05:18,769 --> 00:05:20,049
So they gave us our slope.

96
00:05:20,050 --> 00:05:23,699
It's 3/5, and the y-intercept is negative 3.

97
00:05:23,699 --> 00:05:25,944
So here, immediately, this would be very easy to put it

98
00:05:25,944 --> 00:05:27,420
in the slope-intercept form.

99
00:05:27,420 --> 00:05:35,020


100
00:05:35,019 --> 00:05:40,649
The equation of this line is y is equal to the slope 3/5x,

101
00:05:40,649 --> 00:05:43,199
plus the y-intercept, minus 3.

102
00:05:43,199 --> 00:05:44,069
And we'd be done.

103
00:05:44,069 --> 00:05:47,589
But how do you put this into point-slope form?

104
00:05:47,589 --> 00:05:53,189


105
00:05:53,189 --> 00:05:54,829
Well, we know the slope.

106
00:05:54,829 --> 00:05:56,949
We know that m is equal to 3/5.

107
00:05:56,949 --> 00:05:59,670


108
00:05:59,670 --> 00:06:01,750
But do we know any points on this line?

109
00:06:01,750 --> 00:06:04,149
You need a point and a slope to immediately put it into

110
00:06:04,149 --> 00:06:05,659
point-slope form.

111
00:06:05,660 --> 00:06:07,180
Well, we know one point.

112
00:06:07,180 --> 00:06:08,300
We know the y-intercept.

113
00:06:08,300 --> 00:06:10,150
The y-intercept is negative 3.

114
00:06:10,149 --> 00:06:13,419
That means when x is equal to 0, y is equal to negative 3.

115
00:06:13,420 --> 00:06:16,520
So our point is, the point 0, negative 3.

116
00:06:16,519 --> 00:06:18,120
You could have tried to figure out other points.

117
00:06:18,120 --> 00:06:20,449
You could have said, oh, when x is equal to 5,

118
00:06:20,449 --> 00:06:21,500
y is equal to 0.

119
00:06:21,500 --> 00:06:24,089
There's all sorts of things you could've tried out.

120
00:06:24,089 --> 00:06:26,500
But this one was just sitting there for us.

121
00:06:26,500 --> 00:06:27,839
The y-intercept is negative 3.

122
00:06:27,839 --> 00:06:31,639
That means that the point 0, minus 3 is on the line.

123
00:06:31,639 --> 00:06:34,180
So let's write it in point-slope form.

124
00:06:34,180 --> 00:06:42,150
y minus this y-value, so y minus negative 3, is equal to

125
00:06:42,149 --> 00:06:48,079
the slope 3/5 times x minus this x-value, this

126
00:06:48,079 --> 00:06:52,000
x-coordinate, x minus 0.

127
00:06:52,000 --> 00:06:53,420
And this is point-slope form.

128
00:06:53,420 --> 00:06:58,810
This could be y plus 3 is equal to 3/5 times-- we could

129
00:06:58,810 --> 00:07:00,199
write times x minus 0.

130
00:07:00,199 --> 00:07:03,610
If you really wanted to make it look like point-slope form,

131
00:07:03,610 --> 00:07:05,509
this would be point-slope form, but it's kind of silly

132
00:07:05,509 --> 00:07:07,079
to write x minus 0.

133
00:07:07,079 --> 00:07:09,935
So you could just write y plus 3 is equal to 3/5x.

134
00:07:09,935 --> 00:07:13,289


135
00:07:13,290 --> 00:07:15,379
It's not 100% clear that you're in point-slope

136
00:07:15,379 --> 00:07:16,230
form yet right now.

137
00:07:16,230 --> 00:07:18,480
But I think you still need to write x minus 0.

138
00:07:18,480 --> 00:07:21,870
And obviously, to go from here to there, you just have to

139
00:07:21,870 --> 00:07:24,110
subtract 3 from both sides and you'll get that.

140
00:07:24,110 --> 00:07:25,920
So these are almost equivalent.

141
00:07:25,920 --> 00:07:28,720
I mean, they are equivalent in terms of what they represent.

142
00:07:28,720 --> 00:07:30,750
They're even almost equivalent in how you write them.

143
00:07:30,750 --> 00:07:33,300
You just have to subtract 3 from both sides of this

144
00:07:33,300 --> 00:07:36,350
equation to get to that one.

145
00:07:36,350 --> 00:07:37,770
Let's do another one.

146
00:07:37,769 --> 00:07:42,259
Let's put this in point-slope form.

147
00:07:42,259 --> 00:07:44,879
They're giving us information.

148
00:07:44,879 --> 00:07:48,569
They're giving it in the form of x, f of x.

149
00:07:48,569 --> 00:07:53,680
So in this situation, when x is negative 7, f of

150
00:07:53,680 --> 00:07:55,860
x is equal to 5.

151
00:07:55,860 --> 00:07:59,580
And this coordinate, they're telling us when x is equal to

152
00:07:59,579 --> 00:08:03,810
3, f of x is equal to negative 4.

153
00:08:03,810 --> 00:08:08,220
So just like that, we can figure out the slope first. So

154
00:08:08,220 --> 00:08:14,895
the slope, which is equal change in y over change in x.

155
00:08:14,894 --> 00:08:15,949
So let's do this.

156
00:08:15,949 --> 00:08:29,129
It's negative 4 minus 5, over 3 minus negative 7.

157
00:08:29,129 --> 00:08:32,899
And this is going to be equal to-- negative 4 minus 5 is

158
00:08:32,899 --> 00:08:34,459
negative 9.

159
00:08:34,460 --> 00:08:38,299
And then 3 minus negative 7, that's the same thing as 3

160
00:08:38,298 --> 00:08:40,048
plus 7, that's 10.

161
00:08:40,048 --> 00:08:42,870
So slope is negative 9/10.

162
00:08:42,870 --> 00:08:43,830
And so to put this in point-slope

163
00:08:43,830 --> 00:08:45,083
form is really easy.

164
00:08:45,082 --> 00:08:52,000


165
00:08:52,000 --> 00:08:55,710
It's just going to be y-- let's do it this way-- y

166
00:08:55,710 --> 00:09:01,769
minus-- I'll color code it-- 3 is equal to-- I'll do this

167
00:09:01,769 --> 00:09:06,480
back to the green color-- is equal to the slope, is equal

168
00:09:06,480 --> 00:09:16,414
to negative 9 over 10, times this x minus this coordinate,

169
00:09:16,414 --> 00:09:20,179
x minus negative 4.

170
00:09:20,179 --> 00:09:22,089
Then we can close it.

171
00:09:22,090 --> 00:09:23,560
So this is in point-slope form.

172
00:09:23,559 --> 00:09:26,049
We obviously can simplify this negative a little bit.

173
00:09:26,049 --> 00:09:32,209
We can rewrite it as y minus 3 is equal to negative 9/10

174
00:09:32,210 --> 00:09:35,330
times x, plus 4.

175
00:09:35,330 --> 00:09:37,340
And we are done.

176
00:09:37,340 --> 00:09:38,000