Thank you EP

I have a bag with yellow and blue marbles in it. At the moment, the ratio of blue marbles to yellow marbles is 4:3. If I add 5 blue marbles and remove 3 yellow marbles, the ratio will be 7:3. How many blue marbles were in the bag before I added more?

You can construct two equations from this as follows:

Let b be the original number of blue marbles and y the original number of yellow marbles.

Then  \(\frac{b}{y}=\frac{4}{3}\\\frac{b+5}{y-3}=\frac{7}{3}\)

Solve these simutaneous equations for b.

For polynomials with repeated roots it can be difficult to identify all the factors.  e.g.  y = x² - 4x + 4   (which is y = (x-2)² ).

Thanks for the help!

\(2x - y = a\\ -6x+3y=b\\ \text{Add 3 times equation 1 to equation 2}\\ 0=3a+b\\ b = -3a\\ \dfrac a b = \dfrac{a}{-3a} = -\dfrac 1 3\)

\(\text{From the vertex we have}\\ (y-3) = a(x+3)^2\\ \text{Now from the point}\\ (0-3) = a(-6+3)^2\\ -3 = 9a\\ a=-\dfrac 1 3\\ y = -\dfrac 1 3 (x+3)^2 + 3\)

Thank you, CPhill !

Here's one more way to solve this :

Triangles under the same height have the same area ratio as the ratio of their bases

So

[ BCD] / [ABC] =  10 / 6

[ BCD ] =  [ 10/6] [ ABC]

[BCD ] =  [ 10/6] [ 10]

[BCD ] = [ 5/3 ] [ 10]

[BCD ] =  50 / 3  

Well you should be able to use PEMDAS to figure this out but you want to start by distributing the 4 and the -2 out to the numbers inside the parenthesis first, and then solve for the rest.

but its 20-2(-4+1)
so that gives us 20+8-2 
which simplifies to 26. 

Ah, I see :'') 

Thanks!! 

I don't really know but from the books you have said you liked so far, I think we have a pretty similar taste but i dunno. 

and I will for sure read All the Light We Cannot See, thank you so much CPhill! 

"e"  is the base of the natural log....it is the irrational number 2.718.....

  .

:'( i honestly don't know man... I could not hear an absolute word of what she was saying

hmm, i see...but where does that "\(e^x \)" come from.
what does e represent, does it just represent like an exponential function or something... also what kind of relationship does it have when its compared to slope point because im pretty sure slope point is just like \(y-y_1=m\left(x-x_1\right)\) if im correct. idk. 
 

I don't usually recommend books as I'm never sure about someone's taste....but....you might want to check out  "All The Light We Cannot See"

This might be one of the best books I have read in the last year. Nirvana.....!!!

I don't know what you are referring to.....but.....she might have been pointing out that the  slope of  any tangent line to the function e^x   is   just  the function value itself at the point of tangency

That is....the slope of the tangent line  at  any  point  (a, e^a)  is just   e^a

See the example here  :  https://www.desmos.com/calculator/fzwjzsxk4l

Notice that the slope of the tangent line  at x  = 0  =   e^0  = 1   

believe it or not i was at the book store for 4 hours looking around for books,,,and i ended up with LOTS of choices however my top two favorites were 1984 and Fahrenheit 451 so I was just deciding between those two for like half an hour because they both seemed so interesting to me! 

I ended up choosing 1984, but I am MOST definitely gonna read Farhenheit 451 soon... LOL :) 

and yes I agree Animal Farm is pretty good, I have to say... it's probably one of my favorite books so far! 

Just finished  re-reading "1984" myself.....a real "dystopian" novel

If you like this sort of thing, check out  "Fahrenheit 451"  by  Ray Bradbury

[ "Animal Farm" by Orwell is good, too !!! ......available at Project Gutenberg - Australia free of charge  !!!!]

half life?

thanks Melody! 
I will definitely check them out! including the other books you stated down below as well 

Last one

BD  = x + 29

AD  =  x + 39

AC   =  20

BC  = x + 19

If  AD  =  x + 39   and  BD  = x + 29, then 

AB  =  AD - BD

AB  =  (x + 39) - ( x + 29)

AB  = 10

Then

AB + BC  =  AC

10 + (x + 19)  = 20

29 + x  =  20

x   = -9

Then BC  =  x + 19  =  -9 + 19   =  10

Proof

A        B        C          D

   10       10       10

AC  = 20

BD  = x + 29  =  -9 + 29  =  20

AD =  x + 39  = -9 + 39  = 30

idk but someone does

Thanks, CU   !!!!

9)  AB   = 2x + 30

     BC = 5

     CD  = 8

So

AB  + BC + CD  =  AD

(2x + 30)  + 5 + 8  =  x + 32

2x + 43  = x + 32

x  = 32 - 43

x = -11