CPhill

sin(6x-30)=0.5

This could have infinite answers, but one solution is based on the fact that the sine = 0.5 at 30°.........

6x - 30  = 30    add 30 to both sides

6x = 60            divide both sides by 6

x = 10

[ P.S.....other solutions could be 30, 70, etc.  ]

Let C be the cost of the dryer....and 2C be the cost of the washer.....and we know that :

C + 2C   = 750

3C   = 750    divide both sides by 3

C  = 250   .....cost of the dryer

Remember that we have the property that ....  a^-n  = 1 / aⁿ   ....so, for instance :

5^-2   = 1/ 5²    = 1/ 25

Prime factor each number and line like factors up in columns

60  =  2^2    * 3    * 5

100 = 2^2            * 5^2

75  =              3    * 5^2

120=  2^3   *  3    * 5

From each column, take the factor with the largest power and multiply these together......so we have :

2^3 * 3 * 5^2   = 

8 * 3 * 25 =

24  *  25  =

600

And that's the LCD

x^2 + 7x  =

x ( x + 7 )

And that's it.......!!!

Here might be a slightly easier approach

log₇ (x) + log₇ (x + 6) = 1  

Note that we can write 1 as  log₇(7)

And using a log property, we can write the left side as log₇ [ x * (x + 6)]  ....so we have

log₇ (x * (x + 6) ] =   log₇(7)       and we can get rid of the logs and solve this equation :

x * (x + 6)  = 7

x^2 + 6x  = 7

x^2 +6x - 7  = 0       factor

(x + 7) (x - 1)  =  0      and setting both factors to 0, we have that x = 1, or x = -7

We must reject the second solution because it requires taking the logs of negative numbers in the original equation, and this is not defined........so, the aswer is  .....   x = 1

I think you might mean this:

sin2x/2sinx = cos^2[x\2]-sin^2[x\2]   

                  [  remember,  cos^2x - sin^2x =  cos2x ..... so cos^2(x/2) - sin^2(x/2)  = cosx ]

2sinxcosx / 2sinx    =  cosx

cosx   = cosx

sin 4y= 4sinycosy - 8sin^3ycosy   

             factor out 4sinycosy  on the right

         =  [4sinycosy]  [ 1 - 2sin^2 y]

         =   [2sin2y ] [ cos2y ]

             Remember that sin2y  = 2 siny cosy........so,  2 sin2y cos2y =

            sin 4y

And the right side = the left side

0.75  = 3/4 lb   .....so....

6 / (3/4)  =

6 * (4/3)   =

24/3  =

8  packages

Parallel lines have the same slope, so......

y - (-10)  = (10/7) [x - ( - 6 ) ]     simplify

y + 10 = (10/7) [x + 6 ] 

y  = (10/7)x + 60/7  - 10

y = (10/7)x + 60/7 - 70/7

y = (10/7)x - 10/7      ......and that's in slope-intercept form