1
00:00:00,000 --> 00:00:00,590


2
00:00:00,590 --> 00:00:04,389
What we're going to do in this video is graph some lines

3
00:00:04,389 --> 00:00:08,330
using the x- and y-intercepts of the line.

4
00:00:08,330 --> 00:00:13,060
So first of all, a line is defined by two points.

5
00:00:13,060 --> 00:00:15,760
If you find two points-- if you're able to plot two points

6
00:00:15,759 --> 00:00:18,019
on a line, then you can connect them and keep going in

7
00:00:18,019 --> 00:00:19,050
both directions.

8
00:00:19,050 --> 00:00:21,410
And you will have drawn the line.

9
00:00:21,410 --> 00:00:23,750
What I'm going to do in this video-- the two points I'm

10
00:00:23,750 --> 00:00:30,620
going to pick are the intercepts of the lines.

11
00:00:30,620 --> 00:00:31,380
There's two of them.

12
00:00:31,379 --> 00:00:36,039
There's the x-intercept and then there's the y-intercept.

13
00:00:36,039 --> 00:00:38,509
These are the points where the lines

14
00:00:38,509 --> 00:00:40,859
intersect the x- and y-axes.

15
00:00:40,859 --> 00:00:45,240
And just to make it clear, this right here is the x-axis

16
00:00:45,240 --> 00:00:47,539
on my graph paper.

17
00:00:47,539 --> 00:00:50,089
That right there is my x-axis.

18
00:00:50,090 --> 00:00:52,340
It just keeps going.

19
00:00:52,340 --> 00:00:54,030
That's my x-axis.

20
00:00:54,030 --> 00:00:56,770
This right here, in the middle, going up

21
00:00:56,770 --> 00:00:59,880
and down, is my y-axis.

22
00:00:59,880 --> 00:01:03,844
That right there is my y-axis.

23
00:01:03,844 --> 00:01:05,250
I don't think you can see what I just wrote.

24
00:01:05,250 --> 00:01:07,689
That is the y-axis right there.

25
00:01:07,689 --> 00:01:10,780
So let me show you what I mean by x- and y-intercepts.

26
00:01:10,780 --> 00:01:13,010
So let's do part (a).

27
00:01:13,010 --> 00:01:15,859


28
00:01:15,859 --> 00:01:18,239
I'll do it in a different color, not the same color as

29
00:01:18,239 --> 00:01:19,509
my intercepts.

30
00:01:19,510 --> 00:01:24,609
Part (a), we have the equation y is equal to 2x plus 3.

31
00:01:24,609 --> 00:01:26,549
So first of all, let's think about what happens when x is

32
00:01:26,549 --> 00:01:27,799
equal to 0.

33
00:01:27,799 --> 00:01:30,009


34
00:01:30,010 --> 00:01:32,980
When x is equal to 0, what is y equal to?

35
00:01:32,980 --> 00:01:40,280
y is going to be equal to two times 0 plus 3.

36
00:01:40,280 --> 00:01:42,210
Well, 2 times 0, that's just 0.

37
00:01:42,209 --> 00:01:45,729
So y is equal to 0 plus 3 or y is equal to 3.

38
00:01:45,730 --> 00:01:49,829
So the point x is equal to 0, y is equal to 3, that

39
00:01:49,829 --> 00:01:51,670
satisfies this equation.

40
00:01:51,670 --> 00:01:54,680
Or it's on the line defined by this equation.

41
00:01:54,680 --> 00:01:57,680
So the point 0, 3.

42
00:01:57,680 --> 00:01:58,890
Let me plot that.

43
00:01:58,890 --> 00:02:01,810
The point 0, 3.

44
00:02:01,810 --> 00:02:06,790
Now notice, this point-- this is when x is equal to 0, this

45
00:02:06,790 --> 00:02:09,750
sits on our y-axis.

46
00:02:09,750 --> 00:02:13,099
We call this right here a y-intercept.

47
00:02:13,099 --> 00:02:14,799
It is when x is equal to 0.

48
00:02:14,800 --> 00:02:17,219
But it's called a y-intercept because it sits on the y-axis.

49
00:02:17,219 --> 00:02:21,030
This is where the line intercepts the y-axis.

50
00:02:21,030 --> 00:02:24,330
Now let's do the same for the x-axis.

51
00:02:24,330 --> 00:02:25,830
Let's set y equal to 0.

52
00:02:25,830 --> 00:02:31,020
So if y is equal to 0, we have 0 is equal to 2x plus 3.

53
00:02:31,020 --> 00:02:34,380
We could subtract 3 from both sides of that equation.

54
00:02:34,379 --> 00:02:37,960
And you get negative 3 is equal to 2x.

55
00:02:37,960 --> 00:02:40,189
We can divide both sides by 2.

56
00:02:40,189 --> 00:02:44,680
And you get negative 3/2 is equal to x.

57
00:02:44,680 --> 00:02:49,000
And negative 3/2, that's the same thing as negative 1 1/2.

58
00:02:49,000 --> 00:02:56,050
So we know that the point negative 3/2, 0, right?

59
00:02:56,050 --> 00:02:59,110
The second scenario, y is equal to 0.

60
00:02:59,110 --> 00:03:03,810
We know that this set of values of x and y also satisfy

61
00:03:03,810 --> 00:03:04,449
this equation.

62
00:03:04,449 --> 00:03:06,539
Or that this point also sits on the line.

63
00:03:06,539 --> 00:03:10,179
So negative 3/2-- remember that's the same thing as

64
00:03:10,180 --> 00:03:11,780
negative 1 1/2.

65
00:03:11,780 --> 00:03:14,060
That's right there. y is 0.

66
00:03:14,060 --> 00:03:16,590
So that's that second point we just figured out.

67
00:03:16,590 --> 00:03:19,439
When y is 0, we're dealing with our x-intercept.

68
00:03:19,439 --> 00:03:22,090
We're intercepting the x-axis.

69
00:03:22,090 --> 00:03:25,300
That is the x-intercept right there.

70
00:03:25,300 --> 00:03:28,850
If I wanted to draw the line, it'll look like this.

71
00:03:28,849 --> 00:03:31,319
Very roughly, I'll connect those points and then just

72
00:03:31,319 --> 00:03:33,019
keep going.

73
00:03:33,020 --> 00:03:35,770
Keep going in both directions.

74
00:03:35,770 --> 00:03:38,790
Keep going in both directions forever and then I will have

75
00:03:38,789 --> 00:03:40,870
drawn my line.

76
00:03:40,870 --> 00:03:42,920
Let's do one more of these.

77
00:03:42,919 --> 00:03:45,039
I don't want to make it too messy over here.

78
00:03:45,039 --> 00:03:47,159
Let me clear all this stuff out of the way.

79
00:03:47,159 --> 00:03:50,909


80
00:03:50,909 --> 00:03:54,549
Let me clear this out of the way as well.

81
00:03:54,550 --> 00:03:57,560
You could always pause it, if you want to gaze at it-- oh I

82
00:03:57,560 --> 00:03:59,490
shouldn't have done that, I got rid of my-- well you can

83
00:03:59,490 --> 00:04:01,640
see still the axes.

84
00:04:01,639 --> 00:04:04,439
I'll redraw the axes.

85
00:04:04,439 --> 00:04:06,789
I'll redraw it right there.

86
00:04:06,789 --> 00:04:08,409
Good enough.

87
00:04:08,409 --> 00:04:10,409
That's one, my x-axis.

88
00:04:10,409 --> 00:04:15,879
That is my y-axis right over there.

89
00:04:15,879 --> 00:04:18,110
Let me do part (b). (b) looks like an

90
00:04:18,110 --> 00:04:20,389
interesting one right there.

91
00:04:20,389 --> 00:04:26,860
(b) has 6 times x minus 1 is equal to 2 times y plus 3.

92
00:04:26,860 --> 00:04:30,080
So let's look at the situation when x is equal to zero.

93
00:04:30,079 --> 00:04:34,089
When x is equal to 0, that equation up there becomes 6

94
00:04:34,089 --> 00:04:36,079
times negative 1.

95
00:04:36,079 --> 00:04:36,539
Right?

96
00:04:36,540 --> 00:04:37,590
Because this is 0.

97
00:04:37,589 --> 00:04:42,859
So 6 times negative 1 is negative 6 is equal to 2-- let

98
00:04:42,860 --> 00:04:46,400
me distribute this-- 2y plus 6.

99
00:04:46,399 --> 00:04:49,719
I multiplied 2 times the y and times the 3.

100
00:04:49,720 --> 00:04:53,220
And then we can subtract 6 from both sides, so that you

101
00:04:53,220 --> 00:04:55,080
get negative 12.

102
00:04:55,079 --> 00:04:57,649
Negative 12 is equal to 2y.

103
00:04:57,649 --> 00:05:01,314
I subtracted 6 from both sides to essentially move this 6 on

104
00:05:01,314 --> 00:05:02,779
to the left-hand side.

105
00:05:02,779 --> 00:05:04,449
You subtract 6 here, this disappears.

106
00:05:04,449 --> 00:05:06,909
You subtract 6 from here, you get negative 12.

107
00:05:06,910 --> 00:05:12,020
Divide both sides by 2, you get negative 6 is equal to y.

108
00:05:12,019 --> 00:05:16,109
So our first point that we know is on the line is a point

109
00:05:16,110 --> 00:05:19,300
x is equal to 0, y is equal to negative 6.

110
00:05:19,300 --> 00:05:23,560


111
00:05:23,560 --> 00:05:24,949
That is our y-intercept.

112
00:05:24,949 --> 00:05:28,889


113
00:05:28,889 --> 00:05:34,569
Now let's see what happens when y is equal to 0.

114
00:05:34,569 --> 00:05:35,959
Go back to the original equation.

115
00:05:35,959 --> 00:05:40,329
We have-- let me write that a little bit different-- when y

116
00:05:40,329 --> 00:05:44,259
is equal to 0, we have 6 times x minus 1.

117
00:05:44,259 --> 00:05:45,289
Let me just distribute that.

118
00:05:45,290 --> 00:05:50,220
So 6x minus 6 is equal to 2 times y.

119
00:05:50,220 --> 00:05:51,630
Well, y is just going to be 0.

120
00:05:51,629 --> 00:05:53,430
So it's going to be 2 times 3.

121
00:05:53,430 --> 00:05:54,480
This whole thing is going to be 3.

122
00:05:54,480 --> 00:05:56,550
It's going to be equal to 6.

123
00:05:56,550 --> 00:06:00,310
So let's subtract 6 from both sides-- or let's add 6 to both

124
00:06:00,310 --> 00:06:01,060
sides of this equation.

125
00:06:01,060 --> 00:06:02,800
I want to get rid of that right there.

126
00:06:02,800 --> 00:06:05,379
So I need to add 6 to make that 0.

127
00:06:05,379 --> 00:06:10,000
So we get 6x is equal to-- you add 6 to the right-hand side,

128
00:06:10,000 --> 00:06:11,389
you get 12.

129
00:06:11,389 --> 00:06:15,009
And you get x-- divide both sides by 6, you get

130
00:06:15,009 --> 00:06:16,629
x is equal to 2.

131
00:06:16,629 --> 00:06:21,659
So our other intercept is x is equal to 2, y is equal to 0.

132
00:06:21,660 --> 00:06:25,060
So we have x is equal to 2, y is equal to 0.

133
00:06:25,060 --> 00:06:26,030
Right there.

134
00:06:26,029 --> 00:06:29,299
That is our x-intercept.

135
00:06:29,300 --> 00:06:34,430
And then our actual line, we just connect the dots.

136
00:06:34,430 --> 00:06:35,860
It will look something like that.

137
00:06:35,860 --> 00:06:37,770
Obviously, you have trouble drawing a straight line, but I

138
00:06:37,769 --> 00:06:38,839
think you got the idea.

139
00:06:38,839 --> 00:06:42,379
Two points define a line.

140
00:06:42,379 --> 00:06:44,319
Let me do one more of these.

141
00:06:44,319 --> 00:06:46,870
I could probably do it up here, in this

142
00:06:46,870 --> 00:06:48,220
real estate up here.

143
00:06:48,220 --> 00:06:52,340
Let's do part d.

144
00:06:52,339 --> 00:06:55,359
You can do part c if you want extra practice.

145
00:06:55,360 --> 00:07:00,379
So part d, we have x plus y is equal to 8.

146
00:07:00,379 --> 00:07:02,230
This is actually very, very straightforward.

147
00:07:02,230 --> 00:07:05,500
When x is equal to 0, what is y equal to?

148
00:07:05,500 --> 00:07:09,370
If this is 0, all we have left is y is equal to 8.

149
00:07:09,370 --> 00:07:14,030
And then when y is 0, what is x?

150
00:07:14,029 --> 00:07:15,859
Well, you can just almost cover this up.

151
00:07:15,860 --> 00:07:18,090
Then you just get x plus 0 is equal to 8.

152
00:07:18,089 --> 00:07:20,750
x is equal to 8.

153
00:07:20,750 --> 00:07:24,279
So that was pretty straightforward right there.

154
00:07:24,279 --> 00:07:26,869
So we have one point, 0, 8.

155
00:07:26,870 --> 00:07:28,829
So 0, 8.

156
00:07:28,829 --> 00:07:31,009
That is the y-intercept.

157
00:07:31,009 --> 00:07:33,039
And then we have the point 8, 0.

158
00:07:33,040 --> 00:07:34,060
8, 0.

159
00:07:34,060 --> 00:07:35,470
That is the x-intercept.

160
00:07:35,470 --> 00:07:37,252
And then we can connect the dots.

161
00:07:37,252 --> 00:07:41,460
It looks something like that.

162
00:07:41,459 --> 00:07:44,310
Now let's do this problem down here.

163
00:07:44,310 --> 00:07:48,079
At a local grocery store, strawberries cost $3.00 per

164
00:07:48,079 --> 00:07:52,810
pound and bananas cost $1.00 per pound.

165
00:07:52,810 --> 00:07:59,555
If I have $10 to spend between strawberries and bananas, draw

166
00:07:59,555 --> 00:08:03,139
a graph to show the combination of each I can buy

167
00:08:03,139 --> 00:08:07,319
and spend exactly $10.

168
00:08:07,319 --> 00:08:08,990
Let me draw my axes.

169
00:08:08,990 --> 00:08:12,030
Make sure that these are nice and filled in.

170
00:08:12,029 --> 00:08:14,879
So that remember is my horizontal access.

171
00:08:14,879 --> 00:08:19,620


172
00:08:19,620 --> 00:08:22,069
But just for fun, instead of calling it the x-axis, I'm

173
00:08:22,069 --> 00:08:24,235
going to call it the strawberry-axis.

174
00:08:24,235 --> 00:08:28,280


175
00:08:28,279 --> 00:08:31,639
Actually let me call it s, the s-axis, where s is for

176
00:08:31,639 --> 00:08:33,330
strawberries.

177
00:08:33,330 --> 00:08:38,666
Let s equal the number of strawberries.

178
00:08:38,666 --> 00:08:41,760


179
00:08:41,759 --> 00:08:43,418
I think you know where I'm going.

180
00:08:43,418 --> 00:08:47,829
b will be number of bananas.

181
00:08:47,830 --> 00:08:50,580


182
00:08:50,580 --> 00:08:51,830
Let me do my b-axis.

183
00:08:51,830 --> 00:08:55,340
I'm going to plot the number bananas on the vertical axis.

184
00:08:55,340 --> 00:08:58,759
So this right there, that is the b-axis.

185
00:08:58,759 --> 00:09:00,095
b for bananas.

186
00:09:00,095 --> 00:09:06,070


187
00:09:06,070 --> 00:09:07,860
Strawberries cost $3.00.

188
00:09:07,860 --> 00:09:09,379
OK, this isn't the number of strawberries.

189
00:09:09,379 --> 00:09:10,789
This is the number of pounds.

190
00:09:10,789 --> 00:09:12,039
Let me clear that up.

191
00:09:12,039 --> 00:09:13,429
This is pounds of strawberries.

192
00:09:13,429 --> 00:09:14,989
We're not going by the number of strawberries.

193
00:09:14,990 --> 00:09:16,919
We're going by pounds of strawberries.

194
00:09:16,919 --> 00:09:21,659
This is pounds of bananas.

195
00:09:21,659 --> 00:09:22,679
All right.

196
00:09:22,679 --> 00:09:25,479
Strawberries cost $3.00 per pound, bananas

197
00:09:25,480 --> 00:09:27,080
cost $1.00 per pound.

198
00:09:27,080 --> 00:09:28,800
So how much am I going to spend?

199
00:09:28,799 --> 00:09:33,689
I'm going to spend $3.00 times the number of strawberries

200
00:09:33,690 --> 00:09:35,860
because they're $3.00.

201
00:09:35,860 --> 00:09:38,230
All right, I'm going to spend $3.00 times the number pounds

202
00:09:38,230 --> 00:09:42,750
of strawberries because it's $3.00 per pound plus $1.00

203
00:09:42,750 --> 00:09:45,809
times the number of pounds of bananas.

204
00:09:45,809 --> 00:09:49,329
And they say I have $10.00 to spend on both.

205
00:09:49,330 --> 00:09:52,040
So that's going to be equal to $10.

206
00:09:52,039 --> 00:09:56,009
So 3 times the pounds of strawberries plus the pounds

207
00:09:56,009 --> 00:09:57,649
of bananas are going to be equal to 10.

208
00:09:57,649 --> 00:10:01,709
1b, that's the same thing as b, so I can rewrite this as 3s

209
00:10:01,710 --> 00:10:04,160
plus b is equal to 10.

210
00:10:04,159 --> 00:10:06,610
Now let's plot this.

211
00:10:06,610 --> 00:10:09,235
Let's look at the situation where I get no strawberries.

212
00:10:09,235 --> 00:10:13,029


213
00:10:13,029 --> 00:10:16,120
So my pounds of strawberries are 0.

214
00:10:16,120 --> 00:10:18,279
What does this equation become?

215
00:10:18,279 --> 00:10:22,860
It becomes 3 times 0 plus b is equal to 10.

216
00:10:22,860 --> 00:10:24,700
That's just 0.

217
00:10:24,700 --> 00:10:26,550
So b will be equal to 10.

218
00:10:26,549 --> 00:10:29,299
So I would have the point 0 pounds of strawberries, I

219
00:10:29,299 --> 00:10:31,379
could get 10 pounds of bananas.

220
00:10:31,379 --> 00:10:34,649
So if I get 0 pounds of strawberries, I can get 10

221
00:10:34,649 --> 00:10:38,709
pounds, right there, of bananas.

222
00:10:38,710 --> 00:10:41,509
Now what about the other scenario?

223
00:10:41,509 --> 00:10:44,330
What about the scenario where I get no bananas?

224
00:10:44,330 --> 00:10:46,400
I get 0 pounds of bananas.

225
00:10:46,399 --> 00:10:47,610
Now let's substitute back here.

226
00:10:47,610 --> 00:10:52,120
We have 3 times my pounds of strawberries plus 0 pounds of

227
00:10:52,120 --> 00:10:53,779
bananas will equal 10.

228
00:10:53,779 --> 00:10:55,409
That's just a 0.

229
00:10:55,409 --> 00:10:57,469
Divide both sides by 3.

230
00:10:57,470 --> 00:11:00,960
I could-- if I get no bananas-- I can get 10/3

231
00:11:00,960 --> 00:11:02,400
pounds of strawberries.

232
00:11:02,399 --> 00:11:05,480
Or this is equal to 3 1/3.

233
00:11:05,480 --> 00:11:11,170
So it would have the point 3 1/3 pounds of strawberries, 0

234
00:11:11,169 --> 00:11:12,889
pounds of bananas.

235
00:11:12,889 --> 00:11:15,240
So 3 1/3 pounds of

236
00:11:15,240 --> 00:11:18,289
strawberries, 0 pounds of bananas.

237
00:11:18,289 --> 00:11:21,110
And then we can connect the line.

238
00:11:21,110 --> 00:11:23,009
This whole line I'm going to draw, I am just going to draw

239
00:11:23,009 --> 00:11:24,090
it in the first quadrant.

240
00:11:24,090 --> 00:11:25,879
Because I can't have negative bananas.

241
00:11:25,879 --> 00:11:27,759
I'm not going to sell pounds of bananas.

242
00:11:27,759 --> 00:11:29,210
And I can't have negative strawberries.

243
00:11:29,210 --> 00:11:30,620
We're not talking about selling strawberries.

244
00:11:30,620 --> 00:11:32,810
So I can only buy these things.

245
00:11:32,809 --> 00:11:34,149
This is the amount that I'm buying.

246
00:11:34,149 --> 00:11:38,009
So let me to connect the dots right there.

247
00:11:38,009 --> 00:11:41,700
This is neat because this line shows all of the possible

248
00:11:41,700 --> 00:11:45,580
combinations of pounds of strawberries and bananas.

249
00:11:45,580 --> 00:11:49,700
For example, this point right here-- I don't know if I'm

250
00:11:49,700 --> 00:11:54,360
drawing it exactly-- it looks like if I get about 5 pounds

251
00:11:54,360 --> 00:11:59,200
of bananas, I can get a little under 2 pounds of

252
00:11:59,200 --> 00:12:00,350
strawberries.

253
00:12:00,350 --> 00:12:01,110
That's what that tells me.

254
00:12:01,110 --> 00:12:06,460
If I get 3 pounds of strawberries, if I get exactly

255
00:12:06,460 --> 00:12:10,009
3 pounds of strawberries, I can get 1 pound of bananas.

256
00:12:10,009 --> 00:12:12,720
Every point here represents a combination that

257
00:12:12,720 --> 00:12:16,040
I can get for $10.

258
00:12:16,039 --> 00:12:17,199