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- pooja195

- mathmate

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

- pooja195

ok :)

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- mathmate

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

- pooja195

yes

- 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

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

- pooja195

|dw:1433470740581:dw|

- pooja195

:/

- mathmate

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

- 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

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

- pooja195

|dw:1433471092615:dw|

- pooja195

:/

- mathmate

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

- pooja195

its all solutions greater than y right?

- 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

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

The teacher taught it a diffrent way :/

- pooja195

But similar to that so its ok .-.

- mathmate

|dw:1433471382597:dw|

- mathmate

Can you tell me what he taught?

- pooja195

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

- mathmate

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

- mathmate

* better way

- 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

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

- mathmate

Making sense?

- pooja195

yep

- mathmate

|dw:1433471680888:dw|

- 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

|dw:1433471920376:dw|

- pooja195

/.\

- 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

yes

- mathmate

|dw:1433472096086:dw|

- mathmate

You're pretty good at it!

- mathmate

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

- 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

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

- mathmate

|dw:1433472344866:dw|

- mathmate

That's the kind of response that is sufficient.

- mathmate

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

- 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

o.O im lost .0.

- mathmate

Starting from where?

- pooja195

nvm i understand >_<

- mathmate

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

- mathmate

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

- mathmate

So are we good with 6.8?

- pooja195

yes

- mathmate

We'll do Chapter 7 then.

- pooja195

:)

- mathmate

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

- mathmate

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

- mathmate

|dw:1433473022450:dw|

- mathmate

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

- mathmate

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

- pooja195

yep

- mathmate

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

- mathmate

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

- pooja195

yeesss

- mathmate

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

- 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

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

- mathmate

is that ok so far?

- pooja195

yes

- mathmate

ok, your turn to do something!

- mathmate

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

- mathmate

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

- mathmate

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

- pooja195

|dw:1433473675355:dw|

- pooja195

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

- mathmate

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

- mathmate

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

- pooja195

>:(

- mathmate

|dw:1433474097592:dw|

- mathmate

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

- pooja195

:)

- mathmate

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

- mathmate

* the solution of the system of inequalities.

- mathmate

|dw:1433474286903:dw|

- pooja195

Got it.

- mathmate

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

- pooja195

So shading ? .-.

- mathmate

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

- mathmate

similar to the example I did above.

- pooja195

|dw:1433474544295:dw|

- pooja195

O_O

- pooja195

woah it screwed it up

- mathmate

The solution set is which colour?

- mathmate

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

- pooja195

y>âˆ’x+3=pink
y

- 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

yes

- mathmate

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

- pooja195

T_T

- mathmate

|dw:1433474989120:dw|

- pooja195

|dw:1433475046485:dw|

- pooja195

>:(

- pooja195

It keeps messing it up! >:(

- pooja195

green= y
pinkish=x

- mathmate

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

- pooja195

the pink

- mathmate

the intersection of the two regions!

- mathmate

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

- mathmate

|dw:1433475274853:dw|

- mathmate

still ok?

- pooja195

yea

- mathmate

:( not promising, he's crying!

- mathmate

We only have one more topic to do.

- pooja195

T_T

- mathmate

an easy one. 7.5

- mathmate

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

- mathmate

|dw:1433475491469:dw|

- mathmate

Are we good so far?

- pooja195

i know this...

- mathmate

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

- pooja195

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

- pooja195

two lines interscting...

- 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

Can you rename the three cases?

- mathmate

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

- pooja195

Now its more complicated -_-

- mathmate

|dw:1433475774511:dw|

- pooja195

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

- 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

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

- mathmate

No, I'll give you two exercises now.

- mathmate

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

- pooja195

you know me so well xD

- mathmate

can you elaborate please?

- pooja195

nawh its ok :P

- pooja195

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

- mathmate

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

- pooja195

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

- pooja195

*soltuons

- mathmate

Exactly!

- pooja195

:)

- mathmate

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

- pooja195

:) good

- pooja195

- mathmate

ok, Chapter 8?

- pooja195

yh

- mathmate

gimme a minute to find the page!

- mathmate

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

- mathmate

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

- mathmate

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

- mathmate

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

- pooja195

ok :)

- mathmate

P. 440 gives some vocabularies.

- mathmate

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

- pooja195

A=base

- mathmate

Good!

- pooja195

B= expoenet

- mathmate

Also good!

- mathmate

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

- pooja195

7^5

- mathmate

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

- mathmate

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

- mathmate

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

- pooja195

7^5=16807

- pooja195

-.- nvm

- pooja195

ok :)

- pooja195

i see my mistake .-.

- mathmate

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

- mathmate

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

- pooja195

its ok :0 bai

- mathmate

- mathmate

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

- pooja195

x^4y^5

- mathmate

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

- pooja195

x^1y^2

- mathmate

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

- pooja195

1/xy^2

- mathmate

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

- mathmate

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

- pooja195

^right

- mathmate

Simplify (2x^2)^3

- mathmate

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

- pooja195

8x^6

- mathmate

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

- mathmate

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

- pooja195

kinda :P

- mathmate

please let me know whenever this happens!

- mathmate

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

- pooja195

(âˆ’3)^9=âˆ’19683

- mathmate

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

- mathmate

(-3)^9 is almost good!

- mathmate

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

- 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

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

- pooja195

-_-

- mathmate

That's alright. I am demanding!

- pooja195

-_-

- 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

i havent learned this

- mathmate

Have you learned A=pi r^2 ?

- 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

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

- pooja195

hmm ok....

- mathmate

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

- pooja195

in exponetional form?

- mathmate

Yep! Pretend you have no calculator!

- pooja195

5^8

- mathmate

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

- mathmate

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

- pooja195

125y^12

- mathmate

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

- pooja195

yes
5*5=25*5=125

- mathmate

How about 25*16

- pooja195

400
;-; how many more of these?

- mathmate

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

- mathmate

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

- mathmate

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

- pooja195

-4

- mathmate

Good, still too slow?

- mathmate

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

- mathmate

What is (3pi)^0

- pooja195

1

- mathmate

Excellent!

- mathmate

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

- mathmate

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

- pooja195

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

- mathmate

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

- mathmate

what about things like
solve 4^x=4096

- pooja195

x=6

- mathmate

Calculator or in your head?

- pooja195

-.- paper.

- pooja195

pencil

- mathmate

Excellent! :)

- mathmate

|dw:1433558060131:dw|

- pooja195

we havent learned this

- mathmate

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

- mathmate

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

- pooja195

no ;-;

- mathmate

ok then.

- mathmate

Have you done graphing calculator?

- pooja195

huh?

- mathmate

8.3 is for using the graphing calculator!

- mathmate

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

- pooja195

yea

- mathmate

So you have one too?

- pooja195

ye

- mathmate

Do you know how to graph on the calculator?

- pooja195

no becuse we have never had to

- mathmate

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

- mathmate

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

- mathmate

Guess your teacher does neither! :(

- pooja195

we graph with paper n pencil ;;

- mathmate

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

- pooja195

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

- mathmate

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

- 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

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

- pooja195

5:2