CPhill

consider the line L : kx+3y+12=0 , where k is a constant

i)find k if L is parallel to 6x-y-7=0

ii)find k if L is perpendicular to 6x-y-7=0

Putting the first line in slope-intercept form ,we have

y = (-k/3)x - 4

The second line, in the same form, is :

y = 6x - 7

Since parallel line have the same slope

-k/3   = 6      so k = -18

And perpendicular lines have negative reciprocal slopes, so

3/k   = 6       and k = 1/2

21. A and B can do a piece of work in 15 days; B and C can do it in 12 days while C and A can  finish it 20 days. In how many days will A,B,C finish it working together?

Let A, B, C be their work rates.....and we have.....

15A + 15B = 1   →      60A + 60B = 4    (1)

12B + 12C = 1   →     60B + 60C = 5     (2)

20A + 20C = 1   →     60A  + 60C  = 3    (3)

Subtract equation 3 from equation 1

60B - 60C = 1      add this to equation 2

120B = 6      B = 20 days to do the job

So B does 1/20 of the job in 1 day.....and using equation 1, we have

15A  + 15/20 = 1

15A  = 1/4    A can do 1/60 of the job in one day

And using equation 3, we have

20/60 + 20C = 1

20C  = 2/3

C = 30   so C does 1/30 of the job in one day

So, working together, we have

[1/20 + 1.60 + 1/30]x = 1

[3 + 1 + 2]/ 60  * x = 1

6/60 * x   = 1

1/10 (x)  = 1       x = 10 days  working together

Here's some of them :

1.  711

2. 349 x 10⁶ =   349,000,000

3. sqrt(5929) = 77

4. sqrt(0.8)  =  0.8944271909999159

5. 6.7  = 670%

9. sqrt (11025) = 105

10. cube root (3325) = 15

15.     521 / 1 .08 = P    where P is the original price he paid = about 482.41R

        So.....1.09 x P = about 525.82 R 

16.  (a) 12.23 /  21.43  

       (b) (42x-1 + 10)÷8 = 4     multiply both sides by 8

             42x - 1 + 10   = 32     simplify

             42x + 9   = 32            subtract 9 from both sides

             42x  = 23                   divide both sides by 42

                 x =  23/42

20. solve 9x− 7 (4x+ 2) = 3x+ 4 (4x+ 7)    simplify on both sides

               9x - 28x - 14  =  3x + 16x + 28  combine like terms

                -19x - `14 =   19x + 28        add 14 to both sides, subtract 19x from both sides

                 -38x = 42    divide both sides by -37

                     x = -42/38  =  -21/19

y = 25 - x....so.....

x(25 - x) = 135   simplify

25x - x^2   = 135     rearrange

x^2 - 25x + 135 = 0

The two solutions are   :

[25 + sqrt(85)] / 2     and   [ 25 - sqrt(85)] 2       whichever x is, y is the other.....!!!

Good to see you, TM........!!!!!!

LOL!!!!!!

What's the question  ???

This is just the combinatoric, " 6 choose 3 "  = C(6, 3) =  20 ways

Hey man...you got it..........easy as falling off a log, huh  ???   ....!!!!

3(2s - 4t)   we just distribute the term on the outside over the terms on the inside of the parenthesis....don't forget the signs!!!

And remember

 pos x pos  = pos

neg x neg = pos

pos x neg  or  neg x pos  = neg  

So we have  :

3 x 2s     = 3 x 2 x s   = 6s

And

3 x -4t   =   3 x -4 x t   = -12t

Now....just combine these

6s - 12t

And that's it  !!!!

You know, ipopsie.....I have this theory about math......when you don't see it.....you don't see it.....and when you finally DO see it......you wonder why you had trouble seeing it in the first place  !!!!