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it's a minus sign, it means subtraction

\(4x - (x+8) = \\ 4x - x - 8 = \\ 3x-8\)

Those answers are incorrect. Here is the correct one:

We divide this into cases, the 3 are the same, one is the same as the original and the last 2 are the same, and they are all the same.

Case 1: 5 ways

Case 2: 3*5=15 ways

Case 3: 1 way

Total: 21 ways

total without restrictions:6^3

So probability is 21/216=7/72.

\(\dfrac{5^3}{10^{-2}} = \\ \dfrac{5^3}{(2\cdot 5)^{-2}} = \\ \dfrac{5^3}{2^{-2}5^{-2}} = \\ \dfrac{5^{3-(-2)}}{2^{-2}}=\\ \dfrac{5^5}{2^{-2}} =\\ 5^5 2^2 = \\ 12500\)

Also, it is given that the pair and 3 different already happened, you don't need to calculate that probability.

i got the point thx!!!!!!!!!!!!!!

sorry that was actually a positive 6 in the beginning. I didt write it out that well

But what if the first die out of the 3 is the same as the first 2 that were already rollled, and the last 2 are different. That would work too, right?

-x^2 + 7x^2 = 6x^2

Thank you so much for your information I will use this as a reference when building my own product .

For the last bit,is there anyway you would be able to help me , how would I find the constraints and model the programming ?you dont have to give me the exact answer but just a tip on how I would ? I am sorry I am  confused with these .

I find one

[ x, y]  =  [ 9, 12 ]

x^3 + 5x^2 - 9x - 45       factor by grouping

x^2 ( x + 5)  - 9(x + 5)

(x + 5) ( x^2 - 9)

(x + 5) ( x + 3) ( x - 3)

sqrt ( p + a)  + r  = k       subtract r from both sides

sqrt ( p + a)  =  k  - r      square both sides

p + a  =  ( k - r)^2

p + a  =  k^2 - 2kr + r^2      subtract p from both sides

a  =  k^2  - 2kr   + r^2  - p

From the last problem we saw that a point of interest was a base of 8cm with a volume of 448 cm³

That's where the volumes of the two candle types are equal and thus of equal materials cost to make.

Below this size the brick was cheaper, above it the egyptian is cheaper.

If I had to design a candle I think the following things would be factors

a) taller and slimmer is probably more attractive

b) we want the brick stand securely on the table without tipping so a bigger base is better

c) the egyptian has no tipping problem

So I'd make the brick as big as possible before the volumes became the same, i.e. 8cm.

Larger than that I'd make egyptians and I guess we'd just make them as large as possible i.e. 10cm

cost of the brick is $4.48 + $1.60 = $6.08

50% markup is typical on things like decorative candles so a profit of $3.04

for a total retail price of $9.12

I'd then figure the sales tax for my area.  Here it's 7.5%.  And I'd adjust it for an even dollar value.

A retail price of $9.30 comes to $10 with tax

cost of the egyptian is $7.00 + $1.60 = 8.60$

markup of $4.30 gives a retail price of $12.90

same story with the sales tax.  I'd price it at $13.02 so tax brings it to $14

leaving the last bit for you

2/3  <   4/5 <  6/7

4 b /p     3/2 p/s    5/2 s/c   = 15 b/c      (birds per cow)  

20 b /  15 b/c = 1 1/3 c      ~~~ He will need to trade 2 cows

Let

Chickens  = x

Pigs  = y

Sheep = z

Cows =  w

So

4x = 1y    ⇒  y = 4x 

3y = 2z   ⇒  3(4x)  = 2z ⇒   12x  = 2z  ⇒  6x = z

5z = 2w   ⇒  5(6x)  = 2w  ⇒   30x  = 2w ⇒  15x = 1w

So

1w / 15x   =     ?w / 20x

1 / 15  =  ? / 20

20 / 15  =   ?   =  4/3 cows for 20 chickens

there are 4074 solutions

here's one

(a,b,c,d,e,f,g,h) = {2, 6, 1, 8, 3, 7, 4, 5}

here's another with different digits

(a,b,c,d,e,f,g,h) = {2, 7, 1, 9, 3, 4, 5, 6}

Substitution: x+3y=-4, so x=-4-3y and 3(-4-3y)-7y=36, -12-9y-7y=36, -9y-7y=48, -16y=48, y=-3. and x=x+3(-3)=-4, x-9=-4, x=5. Thus, the ordered pair is (5, -3).

Elimination: We can also show this by elimination: multiply the first equation by 3, which yields, 3x+9y=-12 and 3x-7y=36. Subtract, 16y=-48, y=-3. Plugging the value of y back in to find the value of x, we get x+3(-3)=-4, x-9=-4, x=5. Thus, the ordered pair is (5, -3).

1.   Let's put this equation in to y = mx + b form (easier for me to visualize)

         5x + 8y = -9

           8y = -5x - 9

             y = -5/8  x   - 9/8     slope = -5/8

                 perpindicular slope = 8/5

   We know 10, 10 is on the new line....

      y = 8/5 (x) + b

      10 = (8/5) 10  + b      b = -6

      so     y = 8/5 x  - 6        m + b = -22/5

(2) The perpendicular bisector of the line segment AB is the line that passes through the midpoint of AB and is perpendicular to AB.

Find the equation of the perpendicular bisector of the line segment joining the points (1,2) and (7,4) Enter your answer in the form "y - mx + b."

The perpendicular line will pass through the midpoint  of  (1, 2)  and ( 7,4)

The midpoint is  [   ( 1 + 7)/ 2, (2 + 4) / 2 ]   =  [ 8/2, 6/2 ]  =  ( 4, 3)

The slope between  (1/2) and ( 7, 4)  is   [  4 - 2 ] / [ 7 - 1 ]  =  2 / 6   =  1 / 3

So....the slope of the pepenicular bisector  is   - 3

And the equation of the perpendicular bisector is

y  = - 3 ( x - 4) + 3

y =  -3x + 12 + 3

y = -3x + 15 

The parallel sides of trapezoid ABCD are AD and BC. GIven that sides AB, BC, and CD are each half as long as side AD, find angle D

(1)Let line L1 be the graph of 5x + 8y = -9 . Line L2  is perpendicular to line L1 and passes through the point (10,10). If line L2 is the graph of the equation y = mx + b, then find m + b.

The slope of the line in the form Ax + By  = C   is given by -A / B

So....the line 5x + 8y  = - 9   has the slope  -5/8

A perpendicular line to this one  will have a negative reciprocal slope  =  8/5

So......the equation for L2  with slope 8/5  = m  passing through (10, 10) = ( x1, y1) is

y   = m (x  - x1)  +  y1

y = (8/5) ( x - 10) + 10

y = (8/5)x - 16 + 10

y = (8/5)x  - 6

m =  8/5     and     b   =  -6

So

m +  b  =     8/5 - 6   =   8/5 - 30/5   =   - 22/5

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