CPhill

The first 100 rows of what  ????

[3xy] / [x + y ]  = 5       

[2xz] / [ x + z ]  = 3

[yz ] / [ y + z ]  = 4  ⇒   yz  =  4 [ y + z ]   (1)

3xy  =  5x + 5y   ⇒  3xy - 5y  = 5x  ⇒    y  [ 3x - 5]  = 5x  ⇒  y = [5x] / [ 3x - 5]     (2)

2xz  =  3x + 3z  ⇒   2xz - 3z  = 3x  ⇒    z [ 2x - 3]  = 3x  ⇒   z = [ 3x] / [ 2x - 3]    (3)

Sub  (2)  and (3)  into (1)

[5x] / [ 3x - 5]  *   [ 3x] / [ 2x - 3]  =   4 [  [5x] / [ 3x - 5]   +  [ 3x] / [ 2x - 3]   ]   simplify

15x^2   =     4 [  5x[2x - 3] + 3x [ 3x - 5 ] ]

15x^2  =  4  [ 10x^2 - 15x  +  9x^2 - 15x ]

15x^2  = 76x^2  - 120x

61x^2   -  120x  = 0

x [ 61x - 120] =  0

So  x = 0    [reject as it makes an original denominator  = 0 ]

Or

  61x - 120  = 0   ⇒     x = 120 / 61

So......   z = [ 3x] / [ 2x - 3]  

z =   3 (120 / 61]  /  [  2(120/61)  - 3 ]

z =  [ 360 / 61]  / [ 240/61  - 183/61]

z =  [ 360]  / [ 240 - 183 ]

z =  360 / 57  =     120 / 19

x - 2y  = 0     ⇒  x  = 2y       (1)

y - 2z  = 0    ⇒   y  =  2z  ⇒   y/2  =  z    (2)

x + z  =  5       (3)

Put (1)  and (2)  into  (3)

2y  +  y/2  =  5     multiply through by 2

4y  +  y  =  10

5y  =  10       ⇒   y  =  2

So   x  =  2y   =   4          and    z  =  y/2  =  1

I work cheap.....LOL!!!!

Unfortunately....I can't guarantee that you will pass........!!!!

-135°   =    360  - 135  =  225”

Conversion is

[225 / 180 ] * pi  =

[ 5 / 4 ]   pi       rads

2^25  *  5^27

Note that we can write this in the following manner

[2 * 5 ]^25    *  5^2   =

[10]^25  *   25  =

25  x 10^25  =

2.5 x 10^26       [ in scientific notation  ]

The parabola wll turn upward...it has a minimum value on the defined domain of 1

And the max  on the defined domain  is  2 (2)^2 + 1   =  9 

So......the range on the defined domain is  [ 1 ,9 ]

Here's the graph : https://www.desmos.com/calculator/apgmaihobm

x^2 + 3x + c

To have -2 remain in the range...since this parabola turns upward..it must be that the vertex is at (a, -2)

The x coordinate of the vertex  is given by   -3/  [2 * 1]  =  -3/2

So.....it must be that

(-3/2)^2  + 3(-3/2)  + c  = -2  simplify

9/4  - 9/2  + c  =  -2

-9/4  +  c  =  -2

-9/4  + c  =  -8/4    add 9/4 to both sides

1/4  =  c   ⇒   any larger value takes -2 out of the range

See the graph here : https://www.desmos.com/calculator/b25akxl65z

Anchor Bob and Anna in any of two chairs......and they can be arranged in 2 ways.... [ Bob on Anna's left or right....]

And the other people can be arranged in 7!  = 5040 ways

So....the total arrangements are  2 * 7!  =   10080  ways

There will be two points each for the range values 1 - 9   and one value for 0

So......the max number is  19