chestercat
  • chestercat
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pooja195
  • pooja195
mathmate
  • mathmate
We'll start with 6.8! Graphing linear ineq. in 2 variables!
pooja195
  • pooja195
ok :)

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mathmate
  • mathmate
Are you familiar with plotting one linear equation, such as y=2x+3?
pooja195
  • pooja195
yes
mathmate
  • 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
mathmate
  • mathmate
You know that with a slope of 1, the line is at 45 deg. with the x-axis.
pooja195
  • pooja195
|dw:1433470740581:dw|
pooja195
  • pooja195
:/
mathmate
  • mathmate
Beautiful! I expect much less in details, but that's nice!
mathmate
  • 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.
mathmate
  • mathmate
Can you shade for me the half plane representing y\(\ge\)x+3 ?
pooja195
  • pooja195
|dw:1433471092615:dw|
pooja195
  • pooja195
:/
mathmate
  • mathmate
Very good! How do you decide which one to shade?
pooja195
  • pooja195
its all solutions greater than y right?
mathmate
  • 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.
mathmate
  • 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.
pooja195
  • pooja195
The teacher taught it a diffrent way :/
pooja195
  • pooja195
But similar to that so its ok .-.
mathmate
  • mathmate
|dw:1433471382597:dw|
mathmate
  • mathmate
Can you tell me what he taught?
pooja195
  • pooja195
he said if its greater than to shade up (postivie) If its less than shade down idk something like that >_<
mathmate
  • mathmate
Yes, this is the better that I was going to show you!
mathmate
  • mathmate
* better way
mathmate
  • mathmate
Say if the inequality is y>x+3 check "y>", that means go towards the +(or plus) y-axis, so the upper half.
mathmate
  • mathmate
If it was y<2x+3, then the "y<" tells you to take the half-plane towards the negative y-axis.
mathmate
  • mathmate
Making sense?
pooja195
  • pooja195
yep
mathmate
  • mathmate
|dw:1433471680888:dw|
mathmate
  • 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.
pooja195
  • pooja195
|dw:1433471920376:dw|
pooja195
  • pooja195
/.\
mathmate
  • 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?
pooja195
  • pooja195
yes
mathmate
  • mathmate
|dw:1433472096086:dw|
mathmate
  • mathmate
You're pretty good at it!
mathmate
  • mathmate
Now if we have an inequality x+y\(\le\) 10 Can you draw the line, and the region represented by the inequality?
mathmate
  • 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.
mathmate
  • mathmate
x+y=5 would give an x-intercept of 5 and y-intercept of 5.
mathmate
  • mathmate
|dw:1433472344866:dw|
mathmate
  • mathmate
That's the kind of response that is sufficient.
mathmate
  • mathmate
Both +x and +y are on the negative side, so the region is to the left, and downwards.
mathmate
  • 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.
pooja195
  • pooja195
o.O im lost .0.
mathmate
  • mathmate
Starting from where?
pooja195
  • pooja195
nvm i understand >_<
mathmate
  • mathmate
For x+y<=10 which is in general form, you can convert to y<=-x+10, would that be easier?
mathmate
  • mathmate
One easy one to remember is x+y = K has both intercepts equal to K.|dw:1433472650720:dw|
mathmate
  • mathmate
So are we good with 6.8?
pooja195
  • pooja195
yes
mathmate
  • mathmate
We'll do Chapter 7 then.
pooja195
  • pooja195
:)
mathmate
  • mathmate
System of linear equations and inequalities. ===========================
mathmate
  • mathmate
A system of linear systems is simply a set of lines. With a system of two equations, there are two lines.
mathmate
  • mathmate
|dw:1433473022450:dw|
mathmate
  • mathmate
|dw:1433473082856:dw| represents the system of equations y=x+3 and x+y=8
mathmate
  • mathmate
so far so good, you know where the lines are coming from (how they were drawn)?
pooja195
  • pooja195
yep
mathmate
  • mathmate
The intersection point (5.5, 5.5) represents the solution of the system, x=5.5, y=5.5. Right?
mathmate
  • mathmate
* (2.5,5.5) |dw:1433473201267:dw|
pooja195
  • pooja195
yeesss
mathmate
  • mathmate
Now If we want to solve for the system of inequalities, y\(\le\)x+3 x+y\(\le\)8
mathmate
  • 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)?
mathmate
  • mathmate
|dw:1433473435556:dw| The region which is hatched by both region is our solution set, as shown above.
mathmate
  • mathmate
is that ok so far?
pooja195
  • pooja195
yes
mathmate
  • mathmate
ok, your turn to do something!
mathmate
  • mathmate
Graph and show the solution set (region) of the following system of inequalities: y>-x+3 y
mathmate
  • mathmate
Notice that ">" is represented by a dotted line, since the solution does not include the line.
mathmate
  • mathmate
Also notice that y>-x+3 is the same as x+y>3 (use the two intercepts)
pooja195
  • pooja195
|dw:1433473675355:dw|
pooja195
  • pooja195
its messy this is why i didnt want to do graphs ;-;
mathmate
  • mathmate
It's beautiful! However, it doesn't bother me if you don't use colours, and have no graduations.
mathmate
  • mathmate
Now, remember that the > and < signs exclude the lines from the solution, so must be drawn with dotted lines!
pooja195
  • pooja195
>:(
mathmate
  • mathmate
|dw:1433474097592:dw|
mathmate
  • mathmate
Can you show me how to have the fancy colours (but some other time!)?
pooja195
  • pooja195
:)
mathmate
  • mathmate
Now we need to hatch the region representing the solution. Use hatching (scribble some parallel lines).
mathmate
  • mathmate
* the solution of the system of inequalities.
mathmate
  • mathmate
|dw:1433474286903:dw|
pooja195
  • pooja195
Got it.
mathmate
  • mathmate
k, whenever you're ready, I'm waiting for you to draw/hatch the region of the solution set!
pooja195
  • pooja195
So shading ? .-.
mathmate
  • mathmate
yes, show by shading/hatching the solution set of the system of inequalities.
mathmate
  • mathmate
similar to the example I did above.
pooja195
  • pooja195
|dw:1433474544295:dw|
pooja195
  • pooja195
O_O
pooja195
  • pooja195
woah it screwed it up
mathmate
  • mathmate
The solution set is which colour?
mathmate
  • mathmate
No, I can understand, as long as you specify the colour.
pooja195
  • pooja195
y>−x+3=pink y
mathmate
  • 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?
pooja195
  • pooja195
yes
mathmate
  • mathmate
Here's another system of inequalities, but don't draw the graduations to safe yourself time! x\(\le\)4 y\(\ge\)2
pooja195
  • pooja195
T_T
mathmate
  • mathmate
|dw:1433474989120:dw|
pooja195
  • pooja195
|dw:1433475046485:dw|
pooja195
  • pooja195
>:(
pooja195
  • pooja195
It keeps messing it up! >:(
pooja195
  • pooja195
green= y pinkish=x
mathmate
  • mathmate
That's ok, I can see that they are correct. So the solution set is......
pooja195
  • pooja195
the pink
mathmate
  • mathmate
the intersection of the two regions!
mathmate
  • mathmate
We have two inequalities to be satisfied, so the region must satisfy BOTH regions. I will hatch it below.
mathmate
  • mathmate
|dw:1433475274853:dw|
mathmate
  • mathmate
still ok?
pooja195
  • pooja195
yea
mathmate
  • mathmate
:( not promising, he's crying!
mathmate
  • mathmate
We only have one more topic to do.
pooja195
  • pooja195
T_T
mathmate
  • mathmate
an easy one. 7.5
mathmate
  • mathmate
When we have a system of 2 linear equations, the system is represented by two straight lines, like this.
mathmate
  • mathmate
|dw:1433475491469:dw|
mathmate
  • mathmate
Are we good so far?
pooja195
  • pooja195
i know this...
mathmate
  • mathmate
Do you know what I am about to say? If yes, tell me!
pooja195
  • pooja195
one line = many solutions two lines = one solution parell lines = no solution am i right?
pooja195
  • pooja195
two lines interscting...
mathmate
  • 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:
mathmate
  • mathmate
Can you rename the three cases?
mathmate
  • mathmate
1. non-parallel : _______ solutions 2. parallel and _________ : __________ solutions 3. parallel and __________ : __________ solutions.
pooja195
  • pooja195
Now its more complicated -_-
mathmate
  • mathmate
|dw:1433475774511:dw|
pooja195
  • pooja195
Case 1 = one solution case 2 = no solution case 3 = infinte soltuions
mathmate
  • 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
mathmate
  • mathmate
I'll leave it to you to try some exercise on 7.5
mathmate
  • mathmate
No, I'll give you two exercises now.
mathmate
  • mathmate
Describe the no. of solutions of the following system: x+y=6 2x+2y=12
pooja195
  • pooja195
you know me so well xD
mathmate
  • mathmate
can you elaborate please?
pooja195
  • pooja195
nawh its ok :P
pooja195
  • pooja195
-2(x+y=6) -2x-2y=-12 2x+2y=12 0=0 Infinte solutions.
mathmate
  • mathmate
Very well. now, how many solutions does the following system have: 2x+y=3 y=-2x-6
pooja195
  • pooja195
2x-2x-6+6=3+6 0x=9 0=9 no solution
pooja195
  • pooja195
*soltuons
mathmate
  • mathmate
Exactly!
pooja195
  • pooja195
:)
mathmate
  • mathmate
We're done for 6.8 and 7, how's that?
pooja195
  • pooja195
:) good
pooja195
  • pooja195
mathmate
  • mathmate
ok, Chapter 8?
pooja195
  • pooja195
yh
mathmate
  • mathmate
gimme a minute to find the page!
mathmate
  • mathmate
Ok, in chapter 8, we're doing exponents!
mathmate
  • mathmate
First we need to know the laws of exponents, how to multiply, divide, raise powers, and take roots.
mathmate
  • mathmate
Then an application would be to convert numbers to and from scientific notations.
mathmate
  • mathmate
Finally, there will be a few problems related to decay and growth, e.g. number of rabbits after a certain time, etc.
pooja195
  • pooja195
ok :)
mathmate
  • mathmate
P. 440 gives some vocabularies.
mathmate
  • mathmate
|dw:1433538712158:dw| can you name A and B?
pooja195
  • pooja195
A=base
mathmate
  • mathmate
Good!
pooja195
  • pooja195
B= expoenet
mathmate
  • mathmate
Also good!
mathmate
  • mathmate
Now calculate: (answer in exponential form)|dw:1433538857123:dw|
pooja195
  • pooja195
7^5
mathmate
  • mathmate
Excellent! Can you now write the question AND answer in exponential form?
mathmate
  • mathmate
don't bother with LaTex. We both understand regular text.
mathmate
  • mathmate
I gtg, but answer is 7^2 \(\times\)7^3 = 7^5
pooja195
  • pooja195
7^5=16807
pooja195
  • pooja195
-.- nvm
pooja195
  • pooja195
ok :)
pooja195
  • pooja195
i see my mistake .-.
mathmate
  • mathmate
I'd like it in exponential form! so in general 7^2*7^3 = 7*(2+3) =7^5
mathmate
  • mathmate
or in general x^a * x^b = x^(a+b) this is one of the laws of exponents! sorry gtg
pooja195
  • pooja195
its ok :0 bai
mathmate
  • mathmate
mathmate
  • mathmate
So, what is \(x^3y^2 \times xy^3\). Express answer in exponential form.
pooja195
  • pooja195
x^4y^5
mathmate
  • mathmate
Good, how about \(\large \frac{x^2y^3}{xy}\)
pooja195
  • pooja195
x^1y^2
mathmate
  • mathmate
good! Now express x^3y^2\(\div\)(x^4y^4) in positive exponents.
pooja195
  • pooja195
1/xy^2
mathmate
  • mathmate
Good, I think you mean 1/(xy^2)
mathmate
  • mathmate
Power of a power property: example: (a^2)^3 = a^(2*3) = a^6
pooja195
  • pooja195
^right
mathmate
  • mathmate
Simplify (2x^2)^3
mathmate
  • mathmate
Hint: Product of powers: (a^x b^y)^z = a^(xz)*b^(yz)
pooja195
  • pooja195
8x^6
mathmate
  • mathmate
\(\large (a^x b^y)^z = a^{xz}*b^{yz}\)
mathmate
  • mathmate
Wow, you're good! Am I too slow?
pooja195
  • pooja195
kinda :P
mathmate
  • mathmate
please let me know whenever this happens!
mathmate
  • mathmate
Now simplify ((-3)^3)^3
pooja195
  • pooja195
(−3)^9=−19683
mathmate
  • mathmate
That's a calculator answer! I need the answer in exponential form, if possible!
mathmate
  • mathmate
(-3)^9 is almost good!
mathmate
  • mathmate
I mean it's good, but not simplified enough!
pooja195
  • 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
mathmate
  • mathmate
you can simplify it as (-3)^9=(-1)^9*3^9 = -3^9
pooja195
  • pooja195
-_-
mathmate
  • mathmate
That's alright. I am demanding!
pooja195
  • pooja195
-_-
mathmate
  • mathmate
We have two circles of radii r and 2r. Find the ratio of the area of the big circle to the small circle.
pooja195
  • pooja195
i havent learned this
mathmate
  • mathmate
Have you learned A=pi r^2 ?
pooja195
  • 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
mathmate
  • mathmate
ok, I'll solve this: A1/A2 = pi (2r)^2 /(pi r^2) =(2r)^2/r^2 =4r^2/r^2 =4
pooja195
  • pooja195
hmm ok....
mathmate
  • mathmate
Simplify (-5)^4 * (-5)^4
pooja195
  • pooja195
in exponetional form?
mathmate
  • mathmate
Yep! Pretend you have no calculator!
pooja195
  • pooja195
5^8
mathmate
  • mathmate
Perfect, you got rid of the negative signs because the final power is even! Very good!!
mathmate
  • mathmate
Simplify (5y^2)^3 * (y^3)^2
pooja195
  • pooja195
125y^12
mathmate
  • mathmate
Excellent, did you calculate 125=5^3 in your head?
pooja195
  • pooja195
yes 5*5=25*5=125
mathmate
  • mathmate
How about 25*16
pooja195
  • pooja195
400 ;-; how many more of these?
mathmate
  • mathmate
Just to see if you are used to using your head. My conclusion is that you are! lol
mathmate
  • mathmate
25*16 is the same as counting how much money you have when you have 16 quarters.
mathmate
  • mathmate
What is (1/2)n^3 when n=-2?
pooja195
  • pooja195
-4
mathmate
  • mathmate
Good, still too slow?
mathmate
  • mathmate
Recall: x^0=1, and x^1=x
mathmate
  • mathmate
What is (3pi)^0
pooja195
  • pooja195
1
mathmate
  • mathmate
Excellent!
mathmate
  • mathmate
Rewrite 3x^(-2)/(y^3z^(-1)) with positive exponents!
mathmate
  • mathmate
|dw:1433557420958:dw| What is 'a' on the graph (the y-intercept)
pooja195
  • pooja195
\[\huge\frac{3z}{x^2y^3}\] We did not learning graphing exponetionall functions
mathmate
  • mathmate
Good for the positive exponents, actually, very good! Are you sure about not having done graphics for exponential functions?
mathmate
  • mathmate
what about things like solve 4^x=4096
pooja195
  • pooja195
x=6
mathmate
  • mathmate
Calculator or in your head?
pooja195
  • pooja195
-.- paper.
pooja195
  • pooja195
pencil
mathmate
  • mathmate
Excellent! :)
mathmate
  • mathmate
|dw:1433558060131:dw|
pooja195
  • pooja195
we havent learned this
mathmate
  • mathmate
You see that when a<0, the graph flips about the y-axis.
mathmate
  • mathmate
ik, but just a quick overview, please bear with me!
pooja195
  • pooja195
no ;-;
mathmate
  • mathmate
ok then.
mathmate
  • mathmate
Have you done graphing calculator?
pooja195
  • pooja195
huh?
mathmate
  • mathmate
8.3 is for using the graphing calculator!
mathmate
  • mathmate
Does everybody in school have one? Like the Ti-83 or ti-83+
pooja195
  • pooja195
yea
mathmate
  • mathmate
So you have one too?
pooja195
  • pooja195
ye
mathmate
  • mathmate
Do you know how to graph on the calculator?
pooja195
  • pooja195
no becuse we have never had to
mathmate
  • mathmate
That's good! I am not for it either. It doesn't do much.
mathmate
  • mathmate
It's much easier to teach students how to use the graphics calculator than graphing with pencil and paper.
mathmate
  • mathmate
Guess your teacher does neither! :(
pooja195
  • pooja195
we graph with paper n pencil ;;
mathmate
  • mathmate
But...you just said you didn't graph exponential functions?
pooja195
  • pooja195
Not this im talking about the slope stuff lel this we ahve never done
mathmate
  • mathmate
ok, so you graphed straight lines! Still it's very good that the teacher asked you to graph on pencil and paper.
mathmate
  • 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).
mathmate
  • mathmate
Find the ratio of the weight of a 5-year old cod to the weight of a 2-year-old cod.
pooja195
  • pooja195
5:2