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I reviewed it and realized that the answer was actually $13$ because $2x+10 \equiv 12$ (mod $16$), and because $2x+11$ is one more, we get $12+1 =13$

y  =  [ -sqrt (3)  / 3 [  x

y / x  =   -sqrt (3)  /3

tan θ   =  -sqrt (3)  / 3  =   - 1 /sqrt (3)

arctan (-1/sqrt 3)    = -30°  

Add  180 to this

θ  = 150°   =    5pi/ 6       good job  !!!!

Add them and divide by 2 to get a + b + c = 0. This means a + b = -c. Then, -c = 8, or c = -8. From this, a = 3, b = 5. Then multiply all 3 to get -120

Find all values of z such that 2/z + 5 = 5/z - 4   

Multiply both sides by z to obtain     2 + 5z  =  5 – 4z   

Combine like terms                                9z  =  3  

Divide both sides by 9                             z  =  3/9  =  1/3   

Don't know much about this.....but....maybe   if   x  = 4

n mod 16   =   19 mod 16   =  3

3x  +  7 =   2x  + 11

x   =   4 

3(4)  +  7  = 19  ( mod 16 )    =   3 mod 16

2(4)  + 11  =  19    (mod 16 )   =  3 mod 16

Do you mean 1/16 are yellow?  (You don't get a whole number of balls with the figures you've specified!).

c = 

18

0

- 16

-30

-42

-52

-60

-66

-70

-72

I'll  let you sum them

Also, I've been wondering if the answer is $5$ because when we add $2$ to both sides to get rid of the remainder, we get

1. $3x + 9 \equiv 0$ (mod $16$)

2. $3 \cdot 13 + 9 = 48$ which is divisible by $16$ and when $x = 13$ this is the smallest possible value for $x$ that satisfies this. If we then apply this value to $2x + 11$ we get

3. $2 \cdot 13 + 11 = 37$

4. If we divide $37$ by $16$ we get $2$ r $5$. So, I'm wondering if the answer to this is $5.$

Am I correct?

P.S BTW I'm the same guest.

See EP's  excellent answer here   :   https://web2.0calc.com/questions/percent-question_14

5x^2   - 20x  +  1357   -  80x  +  415          simplify

5x^2  -100x  + 1772

The  minimum  value  willl occur  at x =    100  / ( 2 * 5)   =  10

Put  this  value into the  function  and the  minimum  is

5 (10)^2  -  100 ( 10)  + 1772   =

500  -  1000  + 1772   =

1272

Using  Angle - Side - Angle

Triangle  RQS   is  congruent to   Triangle  QRP

Side - Angle - Angle proves congruence. 

Which triangles have that?

=^._.^=

1. 

6(7) + 2 + 3(5) + 4

2. 

137/500 = 274/1000 = .274

Can you take these from here?

=^._.^=

Check the original post, I've posted the answer there.

a) We have eight slips of 2 and two slips of 9, lets make a probability distribution

E(x) = (2 × 4/5) + (9 × 1/5) = 17/5 = 3.4

b) one additional 9 is added so, 

E(x) = (2 × 8/11) + (9 × 3/11) = 43/11 = 3.9

c) another 9 is added to the bag so, 

E(x) = (2 × 2/3) + (9 × 1/3) = 13/3 = 4.3

d) In order to make the expected value 6,

     Let probability of 2 be x then, probability of 9 is (1-x)

    ⇒ 2x + 9(1-x) = 6

         2x + 9 - 9x = 6

         7x = 3   ⇒x = 3/7

     Probability of 9 = 1- 3/7 = 4/7 

   Let no. of 9 cards added be n then

    \({4+n\over 12+n} = {4\over 7}\)

    28 +7n = 48 + 4n 

     3n  =  20  ⇒ n = 20/3 

e) In order to make the expected value 8,

     Let probability of 2 be x then, probability of 9 is (1-x)

    ⇒ 2x + 9(1-x) = 8

         2x + 9 - 9x = 8

         7x = 1   ⇒x = 1/7

     Probability of 9 = 1- 1/7 = 6/7 

   Let no. of 9 cards added be n then

    \({4+n\over 12+n} = {6\over 7}\)

    24 + 6n = 48 + 4n 

     2n  =  24   ⇒n = 12

∴ '12' 9s  should be added. 

Thank you SOO much!

We have

C (  n ,  k-1)   =                          n!                                      n!

                               ___________________    =       ____________________       (1)

                                ( n - (k -1))! *  ( k -1)!                     ( n - k + 1)!  ( k -1)!

C  (  n , k-2)  =                          n!                                        n !

                                  __________________  =       _____________________       (2) 

                                  ( n - ( k -2))!  ( ( k -2) !              ( n - k + 2)!  ( k - 2)!

Dividing  (1)  by (2)     and  simplifying    we  get

( n - k +2)!   ( k - 2)!

________________    =

( n - k + 1)!   (k - 1)!

( n - k + 2)!                  ( k - 2)  !

___________   *     __________   =

( n - k + 1)!                (   k - 1) !

 n -  k  +  2

__________

     k   -  1

3x + 7 = 2 (mod 16)

3x + 7 = 18 (mod 16)

3x = 11 (mod 16)

3x = 27 (mod 16)

x = 9 (mod 16)

2*9 + 11 = 27 = 11 (mod 16)

=^._.^=

Using the midpoint  formula

(  -2  + x )  / 2   =  1                             ( -1  + y)   / 2   =  -6

-2  +  x   =   2                                     -1  +  y   =   -12

x  =  4                                                 y  =  -11

B  =   (x , y)  =    ( 4 , -11)

Sum   =  -7

2log ( a  -b)  =    log  a   +   log  b

log   ( a - b)^2  =  log  ( ab)

Implies  that

(a - b)^2   =   ab

a^2  -  2ab     +  b^2    =   ab

a^2  +  b^2    =  3ab               divide  through  by ab              

a/b  +  b/a    =   3

a  /  b   =    3  -  b/a

oh I found the answer, I just found f(0) = multiple solutions, split into cases, and some of these cases had more cases, so I just did some mechanical caseowork and found it. BTW your answer was incorrect but thanks for the effort.

We   have the  following  form

x^4   +  bx^3  +  cx^2  +  dx  + e  

We  have  this system

1   -   b    +   c   - d   +  e   =   -1

16  + 8b  + 4c  + 2d  + e  =  -4

81  - 27b  + 9c  - 3d +   e   =   9

256 + 64b  + 16c  + 4d  + e  = -16                rearranging  we  have

-b + c  - d  +  e   =  -2

8b + 4c + 2d  + e  =  -20

-27b + 9c  - 3d  + e  =  -72

64b + 16c  + 4d  + e  = -272

This is tedious to solve  ( but not impossible) so  I'm using some  technology  to  find  b,c,d and e

a = 1

b = -79/35

c  = -89/7

d = 472/35

e  = 768/35

f(1)  is   just  the  sum of these     =  751 /  35

Simple internet search:

isotope = "each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element."

Find AC such that AB= 5sqrt2 and angle b= 45 and angle a=15.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

AD = (5√2) * sin(45º)

AC = AD / cos(30º)

Very nice  , amy   !!!!