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As per given by the question, there are 6 different Algebra books, 5 different Biology books, 4 different Chemistry books, and 3 different copies of the novel Dune.

Now,

(a). 10 books on a shelf if exactly 4 must be Algebra books? 

here, 4 must  be algebra book,

So,

Number of ways to arrange the exactly 4 algebra books are,

4!=244!=24

Remaining box are 6, then there are,

24+6=3024+6=30

(b). all of the books on a shelf if the copies of Dune must be on the ends?

There are 3 different copies of the novel Dune.

Total number of book is 18.

So.

 18p3 = 18/(18-3)!

          = 4896

Hence, 4896 books on a shelf if the copies of Dune must be on the ends.

It is a formula to find the area of a triangle, given the sides. Google has a good explanation of exactly what it is.

Total exams = 200

Undergraduate TAs should get exams to grade =20 

Graduate TAs should get exams to grade = 200 -20 = 180
 

Ways that undergraduate TAs grade the exam =7c5 * 20 =21 * 20 = 420

Ways that graduate TAs grade the exam =7c2 * 180 =21 * 180 

                                                                                   = 3780

Total ways that TAs grade the exam = 7c5 * 20 + 7c2 * 180

                                                               =420 + 3780

                                                               = 4200

Rearrange  as   

y^2  + 5y  - x^2  - x   =  25           complete  the  square  on x  and  y

y^2  + 5y  + 25/4   -  ( x^2  + x  + 1/4)  =  25  + 25/4  - 1/4

(y  + 5/2)^2  -  ( x + 1/2)^2   =  31        divide  through by  31

( y + 5/2)^2             (x + 1/2)^2

_________   -      __________  =      1

       31                         31

This is  a  hyperbola

Rearrange   as

x  -13    =   y^2  -  8y    complete the  square  on  y

Take 1/2 of 8   = 4....square  this  and add to both sides

x  -13 + 16   = y^2  - 8y  +  16       factor  the right side, simplify the left

x  + 3   = ( y - 4)^2

x  =  (y - 4) ^2  -  3

The  vertex  is   (-3, 4)

2x  + y  +  5z   =  4

3x  - 2y +   z =   -17

7x  - 11y + 3z   =  20

Multiply   the  first equation  by  2   =   4x  + 2y  + 10z   = 8

Add  this to the  second equation   =   7x  + 11z   =  -9    ( a)

Multiply the   first  eqation by  11  =   22x  + 11y  + 55z  = 44

Add this to  the third equation  29x  +  58z  =  64     (b)

Multiply (a)  by  29  =   203 x  +  319z  =  -261

Multiply (b)  by - 7   =  -203x  -  406z  = -448

Add the last two equations 

-87z   =  -709

z = 709 / 87

7x +  11(709/87)  =  -9

7x =  -9  - 11 ( 709/87)     =

7x = -8582/87

x = -8582  /( 87 * 7)=  -1226/87 

And

2 (-1226 / 87)  +  y  = 5( 709 / 87)  = 4

y =  4  + 2(1226)/87  - 5 (709) / 87

y = -745  / 87

thank you very much. Appreciate it but what is herons formula?

using herons formula we get that the area of the first triangle is 60. We then plug this into heron's formula again, except with the third side as x this time, and the area. from this we get x = x = +/- 10 and x= +/- 24. X is 24.

3 * 3 * 7 = 63

10  sqrt (3)  = sqrt  300

Using  the  square of  the  distance  formula   we  have

(x - 4)^2  +  ( -7 - 21)^2  =  300      simplify

(x  - 4)^2   + 28^2   = 300

(x - 4)^2  +  784  = 300

(x - 4)^2   =  - 484

No real solution  for  x   ......!!!

3  +  (3/4)x   =  5/6

Multiply  through  by LCM of  4 , 6   =   12

36  + 8x  =  10

Slope   =  [  7  -5 ]  / [ -3 - 5 ]   =  2 / -8   =  -1/ 4

So

[ c - 7 ]  / [ -6 - - 3 ]  =  -1/4

[c - 7 ]  /  [  -3 ]   =  -1/4

c - 7   =  (-1/4) ( -3)

c - 7  =  3/4

c =   3/4 + 7   =   31 / 4

Did you already figure this out? If so, can you tell me what it was?

Ax  + By   = 3                          x +  8y =  -5

A(-7)  + B (-2)  = 3                  A x  + By =  -5

-7A  - 2B  =  3                          The  slope of  this line =  -1/8  = -A/B   ....so....1/8 = A /B

-2B  =  7A  + 3

-2  = (7)(A / B)  + 3/B

-2 =7 ( 1/8)  + 3/B

-2  = (7/8)  +  3/B

- 2 - 7/8  = 3 / B

-23/8  =  3 / B

B =  3  ( 8 / -23)   =  -24/23

A /B  = 1/8

A  / ( -24/23)  =  1/8

A  = (1/8) (-24/23)  = -3/23

So

B   -  A =   (-24/23)  - (-3/23)  =  -21  / 23

If  the   vertex  is     (12, -4)    and   one  of  the  x intercepts  =  (15, 0)

By  symmetry.......the other  x intercept  is  given by    12 -  ( 15 - 12) =  12 - 3   =  9

(n - 2) *  180   >  1970

180n   -  360  >  1970

180n  >  2330

n  =  ceiling   [  2330 /  180  ]   =  ceiling  [ 12.97....  ]   =     13  sides

x +  y   =   1       ⇒  x - 1  =  - y       (1)

x +  z   =  -1      ⇒  x + 1   = - z         (2)

x - y - z =  1       (3)

Sub (1)  and (2)  into  (3)

x + ( x  -1)  + ( x + 1)  =  1

3x = 1

x  =1/3

y  = 2/3

z = -4/3

The area is 36*pi - 14.

There are 488 ways to distribute the bracelets.

Let  the  original  number  of  red apples  =  R    and  the original  number  of green apples =  G

We  have  that

R   +  (2/11)  G   -  (1/3)[ R + (2/11)G ]  =  228

G  - (2/11)G -  (2/5)  [ G -(2/11)G ]     =108

Looking  at  the  second  equation and  simplifying  we  have

(9/11)G  - ( 2/5)G  +  (4/55) G =   108

( 45/55)G  - (22/55)G  + ( 4/55)G    =108

(27/55) G  =108

G =  108  / 27  *  55   =  220  =  original number  of green apples

And  simplifying the  first  equation we  have

R  + (2/11)(220)  -  ( 1/3)  [ R  + (2/11)(220)]  =  228

R  +  40   -  (1/3)R  - (1/3) (40)  = 228

(2/3)R  +  40  - 40/3  =  228

2R  + 120  - 40  =   684

2R  +  80   =  684

2R  =  604

R  =  604  / 2    =  302   =  original number of  red apples

R + G =      302   +  228    =    530

If 3m+4n=47, and m+n=15, what is m?

Hello Guest!

\(3m+4n=47\\ m+n=15\ |\ -m\\ n=15-m\ |\ n\ einsetzen\\ 3m+4(15-m)=47\\ 3m+60-4m=47\\ -m+60=47\ |\ -60 \)

\(-m=47-60\\ -m=-13\ |\ \times (-1)\)

\(m=13\)

  !

log ₃  ( x  -1)  =   -1

In  exponential form we  have  that

3^(-1)  =   x  -1

1/3  = x   -  1

1 + 1/3   = x  =    4/3

nvm  found out it is -19/3 i think