CPhill

Here's the last one....it's could be little tricky, NSS.....but.....turns out....it isn't  !!!

Let  x  be the lbs  of peanuts, y be the lbs of almonds and z be the lbs of raisins

Since we have 11 lbs total, one of the equations is

x  + y +   z  = 11

And we need twice as many lbs of peanuts as almonds

So

2y  = x

Finally......

Price per pound of each type * cost per pound of each type  = total cost

In math terms this gives us

1.50x  + 3.00y  + 1.50z  = 21 

AHA!!....notice, NSS, that we don't really have to do ANY MATH from this point forward......look at the third answer.....it's the only one where the number of pounds of peanuts is twice the number of pounds of almonds .....!!!!!

Let's verify that this is correct

1.50 (6)  + 3.00 (3)  + 1.50 (2)  =

9   + 9   +  3   =

$21  ......!!!!

This just proves that, sometimes, reading the problem and looking at the answers makes more sense than doing a lot of complicated math!!!

2x  - y  + z  =  - 4

              z   =   5

-2x  + 3y  - z  = -10

Since we know that z  = 5......put this into equations one and three and we have

2x - y  + 5  = -4  →   2x  - y   = -9  →    y  = 2x  +  9    (4)

-2x + 3y  -5  = -10  → - 2x + 3y = -5    (5)

Sub  (4)  into (5)  for y   and we have that

-2x + 3 [2x + 9]  = -5

-2x + 6x + 27  = -5    subtract 27 from both sides

4x  = -32       divide both sides by 4

x  = -8

Put this into (4)  to find y

y = 2(-8) + 9

y = -16 + 9

y = -7

So  ...  { x, y, z }  = { -8, -7, 5 }

x +  5y  - 4z  = -10    multiply through by  -2   →   -2x - 10y + 8z  = 20   (1)  

2x - y + 5z  =  -9         (2)

2x  - 10y - 5z  =   0     (3)

Add  (1)  and (2)  ......this gives

-11y  + 13z  =  11      (4)

Add (1)  and (3).....this gives

-20y + 3z  = 20      (5)

Multiply  (4)  by   3   and (5)  by -13   and we have

-33y + 39z  = 33

260y - 39z = -260        add these

227 y  = -227          divide both sides by 227

y  = -1

Subbing this into  (4)  gives that     -11(-1) + 13z = 11   → 11 + 13z  =11 

Subtract 11 from both sides →   13z  = 0         which implies that z  =0

Subbing these values into  (2)   we have that

2x - (-1) + 5(0)  = -9

2x + 1  =  -9

2x  = -10     divide by 2

x  = -5

So  { x, y, z}  =  { -5, -1, 0 }

LOL!!!!!!.....some people might call that "cheating"....I prefer the term, "research"

y  = - √ [ 2x -6]

First.....since we can't take the square root of something < 0....then  2x  - 6  must be ≥ 0

So

2x  - 6 ≥  0

2x ≥ 6

x ≥ 3     and this is the domain  →    [3, infinity )

The range is a little tougher

The  greatest  value that  y can be is when x = 3......which means that y = 0

Note that all x's greater than 3 produce  a larger and larger square root....but...the " - " out front means that  we gett larger and larger negative square roots......so......the range is actually  

( - infinity, 0 ]

See the graph here :  https://www.desmos.com/calculator/0mevks4btb

Note that as x increases, y decreases

I agree with AT.....that's a very nice factoring "trick,"  hectictar.....!!!

4^2x  - 5

This is an exponential.......the domain for an exponential is all real numbers  [ note....2x is defined for all real numbers....so.....4^2x  is, too....because the exponent is just a real number ]

The range is trickier.......if it was just    4^2x  the range would be  (0, infinity)

However.......the " - 5 "   part  shifts the graph down by 5 units.....so.....the range now becomes

(-5, infinity)

See the graph here :  https://www.desmos.com/calculator/szmhu8kwjm

Notice that graph almost touches   y = -5    [ but not quite  !!!]

We need to find the variation constant, k

So we have

P  = kA

8160  = 8k   divide both sides by 8

1020   = k

So

P =  1020 * 12   =  12,240 watts

This is a binomial probability and is given by

C(5,4) (2/3)^4 (1/3)  = 80 /243  ≈ 32.9%

x + y + z  = 0   →-  -(x + y)  = z  →   z^3 =  - (x^3 + 3x^2y + 3xy^2 + y^3)

xyz  = 17

xy [ -(x + y) ]  = 17

xy (x + y)  = -17  →  x^2y + xy^2  = -17  →  3x^2y + 3xy^2  = -51

So......

x^3  + y^3  +  [ z^3 ]

x^3  + y^3  +  [ -  ( x^3 + 3x^2y + 3xy^2 + y^3) ]  =

-3x^2y - 3xy^2  =

-[ 3x^2y + 3xy^2]  =

- [-51]  =

51