geno3141

This morning:

     James had  x  dollars

     Brother had  2x  dollars (so James had half as much as his brother)

     Sister had  x + 27 dollars (so James had 27 dollars less than his sister).

His brother paid  12  dollars to James; now James has  x + 12  dollars.

His brother paid  12  dollars to his sister; now his sister has  x + 27 + 12  =  x + 39  dollars.

After paying  12  dollars to both James and his sister, his brother now has  2x - 12 - 12  =  2x - 24  dollars.

James sister now has twice as much money as his brother:  sister  =  2 x brother     --->   x + 39  =  2(2x - 24)

     --->     x + 39  =  4x - 48

     --->     39 + 48  =  4x - x

     --->     87  =  3x

     --->     x  =  29

James now has $29.00 + $12,00  =  $41.00. 

2x + -2(2/3x + 1/3)  =  2

Distribute the -2:

2x + -2(2/3x) + -2(1/3)  =  2

Simplify:

2x -4/3x - 2/3  =  2

Multiply each term by 3 to remove denominators:

3[2x - 4/3x - 2/3]  =  2[3]

Distribute the 3:

3(2x) - (3)(4/3x) - (3)(2/3)  =  6

Simplify:

6x - 4x - 2  =  6

       2x - 2  =  6

             2x  =  8

               x  =  4

"cosh" is hyperbolic cosine.

The hyperbolic functions are based upon a hyperbola and not a circle.

A good explanation can be found in Wikipedia:  https://en.wikipedia.org/wiki/Hyperbolic_function

I'm not certain exactly what you mean: let me try this:

Since  35  =  5 x 7,  log(35)  =  log(5) + log(7);

so  35  =  10^{log(5) + log(7)}.

If a 60 inch long string forms a border around a square picture, each side of the picture must be 60 / 4  = 15 inches long.

To find the area of the picture, multiply the length times the width:  15 x 15  =  225 square inches.

The ratio is:  M/R

If this ratio is the original ratio, not reduced, this would be correct:  M + R  =  K;  so  R  =  K - M.

However, if this is a reduced ratio, then we need to find a value x such that  M/R  =  Mx/Rx  and  Mx + Rx  =  K.

Solving this, we get:  Rx  =  K - Mx  and  R  =  (K - Mx) / x.

w(-4 + z)  =  mz + 17

-4w + wz  =  mz + 17                                (using the distributive law)

wz - mz  =  17 + 4w                                  (subtracting mz from both sides; adding 4w to both sides)

(w - m)z  =  17 + 4w                                 (factoring)

z  =  (17 + 4w) / (w - m)                            (dividing both sides by w - m)

log₅(x) + log₅(2x + 3)  =  1

Since logs are added when the original problem was multiplied, going backwards we have:

log₅[ x · (2x + 3) ]  =  1

log₅( 2x² + 3x )  =  1

Writing the log in exponential notation:

2x² + 3x  =  5¹

--->   2x² + 3x  =  5     --->     2x² + 3x - 5  =  0     --->     (2x + 5)(x - 1)  =  0

                                                                         --->     Either  2x + 5  =  0     or     x - 1  = 0

                                                                         --->     Either     x  =  - 5/2     or     x  =  1

Since  log₅(x)  is undefined if x is a negative number, the value  -5/2  must be ignored; thus the answer is  x  =  1

3x + y  =  -2

5x + 2y  =  -5

I'm choosing to eliminate y from both equations by multiplying the top equation by -2 and then adding down the columns.

-2[3x + y  =  -2]     --->     -6x - 2y  =   4

                                        5x + 2y  =  -5

                              --->     -x         =  -1     --->     x = 1

Replacing x with 1 in the top equation:  3(1) + y  =  -2

                                                                    3 + y  =  -2

                                                                          y  =  -5

Since cos  =  1/sec, when  sec = 2,  cos  =  12