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## pooja195 one year ago @mathmate

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1. pooja195

@mathmate

2. mathmate

We'll start with 6.8! Graphing linear ineq. in 2 variables!

3. pooja195

ok :)

4. mathmate

Are you familiar with plotting one linear equation, such as y=2x+3?

5. pooja195

yes

6. mathmate

In the slope-intercept form, y=ax+b, b is the y-intercept. Ok, good! Can you plot (draw) for me y=x+3

7. mathmate

You know that with a slope of 1, the line is at 45 deg. with the x-axis.

8. pooja195

|dw:1433470740581:dw|

9. pooja195

:/

10. mathmate

Beautiful! I expect much less in details, but that's nice!

11. mathmate

The line divides the x-y plane into two "half-planes". Each half plane represents the solution to the inequality of the form y$$\ge$$x+3 or y$$\le$$x+3.

12. mathmate

Can you shade for me the half plane representing y$$\ge$$x+3 ?

13. pooja195

|dw:1433471092615:dw|

14. pooja195

:/

15. mathmate

Very good! How do you decide which one to shade?

16. pooja195

its all solutions greater than y right?

17. mathmate

The book tells you to test a point, and see if the point is in the good region. If yes, you shade the region containing the point. If not, you shade the other region.

18. mathmate

For example, we can use (0,0) as a test point. (0,0) is below the line. We check if 0$$\ge$$0+3 is true. Since it is not true, (0,0) is not in the region, so it must be the "other" region, the one above the line.

19. pooja195

The teacher taught it a diffrent way :/

20. pooja195

But similar to that so its ok .-.

21. mathmate

|dw:1433471382597:dw|

22. mathmate

Can you tell me what he taught?

23. pooja195

he said if its greater than to shade up (postivie) If its less than shade down idk something like that >_<

24. mathmate

Yes, this is the better that I was going to show you!

25. mathmate

* better way

26. mathmate

Say if the inequality is y>x+3 check "y>", that means go towards the +(or plus) y-axis, so the upper half.

27. mathmate

If it was y<2x+3, then the "y<" tells you to take the half-plane towards the negative y-axis.

28. mathmate

Making sense?

29. pooja195

yep

30. mathmate

|dw:1433471680888:dw|

31. mathmate

Here are three examples, use your teacher's method to shade the correct half-plane. I forgot to write $$\le$$ or $$\ge$$ for the first two cases.

32. pooja195

|dw:1433471920376:dw|

33. pooja195

/.\

34. mathmate

The first (top-left) and the third (on the right) are correct. The second asks for y>-x/3-2, so the "y>" tells us to go for the + y-axis, so the region is above the line. Is that ok so far?

35. pooja195

yes

36. mathmate

|dw:1433472096086:dw|

37. mathmate

You're pretty good at it!

38. mathmate

Now if we have an inequality x+y$$\le$$ 10 Can you draw the line, and the region represented by the inequality?

39. mathmate

When we have the general form of the equation, we can find the y-intercept by setting x=0, and solve for y. Similarly for the x-intercept (zero), we set y=0 and solve for x.

40. mathmate

x+y=5 would give an x-intercept of 5 and y-intercept of 5.

41. mathmate

|dw:1433472344866:dw|

42. mathmate

That's the kind of response that is sufficient.

43. mathmate

Both +x and +y are on the negative side, so the region is to the left, and downwards.

44. mathmate

It may look difficult because it is given in the general form. You can convert the general form to the point-slope form with which you are familiar, or use the intercepts I showed above.

45. pooja195

o.O im lost .0.

46. mathmate

Starting from where?

47. pooja195

nvm i understand >_<

48. mathmate

For x+y<=10 which is in general form, you can convert to y<=-x+10, would that be easier?

49. mathmate

One easy one to remember is x+y = K has both intercepts equal to K.|dw:1433472650720:dw|

50. mathmate

So are we good with 6.8?

51. pooja195

yes

52. mathmate

We'll do Chapter 7 then.

53. pooja195

:)

54. mathmate

System of linear equations and inequalities. ===========================

55. mathmate

A system of linear systems is simply a set of lines. With a system of two equations, there are two lines.

56. mathmate

|dw:1433473022450:dw|

57. mathmate

|dw:1433473082856:dw| represents the system of equations y=x+3 and x+y=8

58. mathmate

so far so good, you know where the lines are coming from (how they were drawn)?

59. pooja195

yep

60. mathmate

The intersection point (5.5, 5.5) represents the solution of the system, x=5.5, y=5.5. Right?

61. mathmate

* (2.5,5.5) |dw:1433473201267:dw|

62. pooja195

yeesss

63. mathmate

Now If we want to solve for the system of inequalities, y$$\le$$x+3 x+y$$\le$$8

64. mathmate

The boundaries of the required region are shown by the lines on the graph. Can you indicate the region (by "hatching" the region with parallel lines)?

65. mathmate

|dw:1433473435556:dw| The region which is hatched by both region is our solution set, as shown above.

66. mathmate

is that ok so far?

67. pooja195

yes

68. mathmate

ok, your turn to do something!

69. mathmate

Graph and show the solution set (region) of the following system of inequalities: y>-x+3 y<x+1

70. mathmate

Notice that ">" is represented by a dotted line, since the solution does not include the line.

71. mathmate

Also notice that y>-x+3 is the same as x+y>3 (use the two intercepts)

72. pooja195

|dw:1433473675355:dw|

73. pooja195

its messy this is why i didnt want to do graphs ;-;

74. mathmate

It's beautiful! However, it doesn't bother me if you don't use colours, and have no graduations.

75. mathmate

Now, remember that the > and < signs exclude the lines from the solution, so must be drawn with dotted lines!

76. pooja195

>:(

77. mathmate

|dw:1433474097592:dw|

78. mathmate

Can you show me how to have the fancy colours (but some other time!)?

79. pooja195

:)

80. mathmate

Now we need to hatch the region representing the solution. Use hatching (scribble some parallel lines).

81. mathmate

* the solution of the system of inequalities.

82. mathmate

|dw:1433474286903:dw|

83. pooja195

Got it.

84. mathmate

k, whenever you're ready, I'm waiting for you to draw/hatch the region of the solution set!

85. pooja195

So shading ? .-.

86. mathmate

yes, show by shading/hatching the solution set of the system of inequalities.

87. mathmate

similar to the example I did above.

88. pooja195

|dw:1433474544295:dw|

89. pooja195

O_O

90. pooja195

woah it screwed it up

91. mathmate

The solution set is which colour?

92. mathmate

No, I can understand, as long as you specify the colour.

93. pooja195

y>−x+3=pink y<x+1= mint

94. mathmate

For both of them to be true, the region is blue (intersection of pink and mint). So the blue region is the solution set. Is that ok?

95. pooja195

yes

96. mathmate

Here's another system of inequalities, but don't draw the graduations to safe yourself time! x$$\le$$4 y$$\ge$$2

97. pooja195

T_T

98. mathmate

|dw:1433474989120:dw|

99. pooja195

|dw:1433475046485:dw|

100. pooja195

>:(

101. pooja195

It keeps messing it up! >:(

102. pooja195

green= y pinkish=x

103. mathmate

That's ok, I can see that they are correct. So the solution set is......

104. pooja195

the pink

105. mathmate

the intersection of the two regions!

106. mathmate

We have two inequalities to be satisfied, so the region must satisfy BOTH regions. I will hatch it below.

107. mathmate

|dw:1433475274853:dw|

108. mathmate

still ok?

109. pooja195

yea

110. mathmate

:( not promising, he's crying!

111. mathmate

We only have one more topic to do.

112. pooja195

T_T

113. mathmate

an easy one. 7.5

114. mathmate

When we have a system of 2 linear equations, the system is represented by two straight lines, like this.

115. mathmate

|dw:1433475491469:dw|

116. mathmate

Are we good so far?

117. pooja195

i know this...

118. mathmate

Do you know what I am about to say? If yes, tell me!

119. pooja195

one line = many solutions two lines = one solution parell lines = no solution am i right?

120. pooja195

two lines interscting...

121. mathmate

You know the topic, but your list is not very precise. We are working with 2 lines. The intersection (or lack of it) can be described in three cases:

122. mathmate

Can you rename the three cases?

123. mathmate

1. non-parallel : _______ solutions 2. parallel and _________ : __________ solutions 3. parallel and __________ : __________ solutions.

124. pooja195

Now its more complicated -_-

125. mathmate

|dw:1433475774511:dw|

126. pooja195

Case 1 = one solution case 2 = no solution case 3 = infinte soltuions

127. mathmate

Exactly! 1. interscting, one solution eg. 2x+3y=4, 4x-2y=0 2. parallel and non-coincident: no solution eg. x+y=10, y=-x+7 3. parallel and coincident: infinite solutions. 2x+2y=4, y=-x+2

128. mathmate

I'll leave it to you to try some exercise on 7.5

129. mathmate

No, I'll give you two exercises now.

130. mathmate

Describe the no. of solutions of the following system: x+y=6 2x+2y=12

131. pooja195

you know me so well xD

132. mathmate

can you elaborate please?

133. pooja195

nawh its ok :P

134. pooja195

-2(x+y=6) -2x-2y=-12 2x+2y=12 0=0 Infinte solutions.

135. mathmate

Very well. now, how many solutions does the following system have: 2x+y=3 y=-2x-6

136. pooja195

2x-2x-6+6=3+6 0x=9 0=9 no solution

137. pooja195

*soltuons

138. mathmate

Exactly!

139. pooja195

:)

140. mathmate

We're done for 6.8 and 7, how's that?

141. pooja195

:) good

142. pooja195

@mathmate

143. mathmate

ok, Chapter 8?

144. pooja195

yh

145. mathmate

gimme a minute to find the page!

146. mathmate

Ok, in chapter 8, we're doing exponents!

147. mathmate

First we need to know the laws of exponents, how to multiply, divide, raise powers, and take roots.

148. mathmate

Then an application would be to convert numbers to and from scientific notations.

149. mathmate

Finally, there will be a few problems related to decay and growth, e.g. number of rabbits after a certain time, etc.

150. pooja195

ok :)

151. mathmate

P. 440 gives some vocabularies.

152. mathmate

|dw:1433538712158:dw| can you name A and B?

153. pooja195

A=base

154. mathmate

Good!

155. pooja195

B= expoenet

156. mathmate

Also good!

157. mathmate

Now calculate: (answer in exponential form)|dw:1433538857123:dw|

158. pooja195

7^5

159. mathmate

Excellent! Can you now write the question AND answer in exponential form?

160. mathmate

don't bother with LaTex. We both understand regular text.

161. mathmate

I gtg, but answer is 7^2 $$\times$$7^3 = 7^5

162. pooja195

7^5=16807

163. pooja195

-.- nvm

164. pooja195

ok :)

165. pooja195

i see my mistake .-.

166. mathmate

I'd like it in exponential form! so in general 7^2*7^3 = 7*(2+3) =7^5

167. mathmate

or in general x^a * x^b = x^(a+b) this is one of the laws of exponents! sorry gtg

168. pooja195

its ok :0 bai

169. mathmate

@pooja195

170. mathmate

So, what is $$x^3y^2 \times xy^3$$. Express answer in exponential form.

171. pooja195

x^4y^5

172. mathmate

Good, how about $$\large \frac{x^2y^3}{xy}$$

173. pooja195

x^1y^2

174. mathmate

good! Now express x^3y^2$$\div$$(x^4y^4) in positive exponents.

175. pooja195

1/xy^2

176. mathmate

Good, I think you mean 1/(xy^2)

177. mathmate

Power of a power property: example: (a^2)^3 = a^(2*3) = a^6

178. pooja195

^right

179. mathmate

Simplify (2x^2)^3

180. mathmate

Hint: Product of powers: (a^x b^y)^z = a^(xz)*b^(yz)

181. pooja195

8x^6

182. mathmate

$$\large (a^x b^y)^z = a^{xz}*b^{yz}$$

183. mathmate

Wow, you're good! Am I too slow?

184. pooja195

kinda :P

185. mathmate

please let me know whenever this happens!

186. mathmate

Now simplify ((-3)^3)^3

187. pooja195

(−3)^9=−19683

188. mathmate

That's a calculator answer! I need the answer in exponential form, if possible!

189. mathmate

(-3)^9 is almost good!

190. mathmate

I mean it's good, but not simplified enough!

191. pooja195

o.O i used the calc im allowed to use on the finals .-. this is what i got what more simplifed can it be? o.O

192. mathmate

you can simplify it as (-3)^9=(-1)^9*3^9 = -3^9

193. pooja195

-_-

194. mathmate

That's alright. I am demanding!

195. pooja195

-_-

196. mathmate

We have two circles of radii r and 2r. Find the ratio of the area of the big circle to the small circle.

197. pooja195

i havent learned this

198. mathmate

Have you learned A=pi r^2 ?

199. pooja195

We havent done much with that i know that stuff from last year though (formula) idk that nor have we learned it THIS year

200. mathmate

ok, I'll solve this: A1/A2 = pi (2r)^2 /(pi r^2) =(2r)^2/r^2 =4r^2/r^2 =4

201. pooja195

hmm ok....

202. mathmate

Simplify (-5)^4 * (-5)^4

203. pooja195

in exponetional form?

204. mathmate

Yep! Pretend you have no calculator!

205. pooja195

5^8

206. mathmate

Perfect, you got rid of the negative signs because the final power is even! Very good!!

207. mathmate

Simplify (5y^2)^3 * (y^3)^2

208. pooja195

125y^12

209. mathmate

Excellent, did you calculate 125=5^3 in your head?

210. pooja195

yes 5*5=25*5=125

211. mathmate

How about 25*16

212. pooja195

400 ;-; how many more of these?

213. mathmate

Just to see if you are used to using your head. My conclusion is that you are! lol

214. mathmate

25*16 is the same as counting how much money you have when you have 16 quarters.

215. mathmate

What is (1/2)n^3 when n=-2?

216. pooja195

-4

217. mathmate

Good, still too slow?

218. mathmate

Recall: x^0=1, and x^1=x

219. mathmate

What is (3pi)^0

220. pooja195

1

221. mathmate

Excellent!

222. mathmate

Rewrite 3x^(-2)/(y^3z^(-1)) with positive exponents!

223. mathmate

|dw:1433557420958:dw| What is 'a' on the graph (the y-intercept)

224. pooja195

$\huge\frac{3z}{x^2y^3}$ We did not learning graphing exponetionall functions

225. mathmate

Good for the positive exponents, actually, very good! Are you sure about not having done graphics for exponential functions?

226. mathmate

what about things like solve 4^x=4096

227. pooja195

x=6

228. mathmate

Calculator or in your head?

229. pooja195

-.- paper.

230. pooja195

pencil

231. mathmate

Excellent! :)

232. mathmate

|dw:1433558060131:dw|

233. pooja195

we havent learned this

234. mathmate

You see that when a<0, the graph flips about the y-axis.

235. mathmate

ik, but just a quick overview, please bear with me!

236. pooja195

no ;-;

237. mathmate

ok then.

238. mathmate

Have you done graphing calculator?

239. pooja195

huh?

240. mathmate

8.3 is for using the graphing calculator!

241. mathmate

Does everybody in school have one? Like the Ti-83 or ti-83+

242. pooja195

yea

243. mathmate

So you have one too?

244. pooja195

ye

245. mathmate

Do you know how to graph on the calculator?

246. pooja195

no becuse we have never had to

247. mathmate

That's good! I am not for it either. It doesn't do much.

248. mathmate

It's much easier to teach students how to use the graphics calculator than graphing with pencil and paper.

249. mathmate

Guess your teacher does neither! :(

250. pooja195

we graph with paper n pencil ;;

251. mathmate

But...you just said you didn't graph exponential functions?

252. pooja195

Not this im talking about the slope stuff lel this we ahve never done

253. mathmate

ok, so you graphed straight lines! Still it's very good that the teacher asked you to graph on pencil and paper.

254. mathmate

ok, "the average weight w (in pounds) of an Atlantic cod can be modelled by w=1.21(1.42)t where t is the age of the fish(in years).

255. mathmate

Find the ratio of the weight of a 5-year old cod to the weight of a 2-year-old cod.

256. pooja195

5:2