CPhill

54 is correct   !!!

Multiply the bottom  equation through by 3

This gives the system

6x +  5y  = 2

-6x + 21y = 102       add these

26y  =  104       divide both sides by 26

y = 4

And

6x + 5(4)  = 2

6x + 20 = 2        subtract 20 from both sides

6x  = -18     divide both sides by 6

x = -3

So   { x, y}  =  { -3, 4 }

Multiply the first equation by  3  ⇒   12x -3y   =  9

Add this to the second equation and we get

14x = 28      divide both sides by 14

x = 2

And using  4(2) - y  = 3   ⇒   8 - y  = 3 ⇒    y  = 5

So

{x, y }  = { 2 , 5 }

The mouse will be closest to the cheese at the point where a perpendicular line through (12,10) intersects the given line 

The slope of the given iine  is -5x....so....the slope of a perpendicular line  = 1/5

And we can write the equation of this line passing through  (12,10)

So  we have

y = (1/5) (x - 12) + 10

y = (1/5)x -12/5 + 10

y = (1/5)x + 38/5....   find the x coordinate of the point we seek by setting the lines equal

(1/5)x + 38/5   =  -5x + 18

5x + (1/5)x  = 18 - 38/5

(26/5)x = 52/5

26x  = 52

x  = 2

And y  =  -5(2) + 18  = 8

So....(a, b)  =  (2,8)     and  a + b  =  10

See the graph here :  https://www.desmos.com/calculator/hb1g4nnou4

The smallest book can fit in any of three positions....and all tthe rest can fit in 3! ways.....

So

3 * 3!  =     3 * 6   =   18 ways

2.       8 cos^4 x - 10 cos^2 x + 3 = 0

(4cos^2x - 3) ( 2cos^2x - 1)  = 0

4cos^2x - 3  = 0

4cos^2x = 3

cos^2x = 3/4

cosx = √3/2       and this happens at  x  =  30°   and x = 330°

cos x = - √3/2    and this happens at  x = 150°   and x  =  210°

2cos^2x - 1  = 0

2cos^2x  = 1

cos^2 x  = 1/2

cos x = 1/√2      and this happens at x = 45°  and x = 315°

cos x = -1  /√2    and this happens at x =135°  and x = 225°

LOL!!!!!

1.  sin x cos x + 2 cos x = 3 sin x + 6

sinxcosx + 2cos x - 3sinx - 6  = 0

(cosx -3) (sin x + 2)  = 0

No solutions    cosx  - 3  = 0   implies that   cos x  = 3   [impossible]

And       sin x + 2  = 0  implies that   sin x  = - 2   [ also impossible ]

Don't remember seeing a user by either of those names....don't see them, either.....yep.....if you don't have too many points and you are inactive for awhile.....I think you disappear into the Cornfield.....!!!!  

OK....ML....!!!!