geno3141

5x - y  =  54    --->     5x  -   y  =   54

3y  =  5x - 72   --->   -5x + 3y  =  -72

Add down:                          2y  =  -18

Divide by 2:                          y  =  -9

Substitue this back into either equation to get the value for x.

There will be one, and only one, solution. 

You can find several by searching for "order of operation video".

The formula for combinations is a modification of the formula for permutations.

The permutation formula:  _nP_r  =  n! / (n - r)!

With permutations, order is important; with combinations, order is not important.

With permutations, these sets of three letters are different:  ABC, ACB, BAC, BCA, CAB, CBA.

With combinations, these sets are not different, because their order is not important.

So these 6 sets in a permutation create only 1 set in a combination.

Thus, you have to divide the answer in the permutation by the number of ways that the selected letters can be arranged.

The number of ways that r elements can be arranged is r!.

So divide the permutation answer by r! to get the combinations answer:  _nC_r  =  n! / [ (n - r)! · r! ]

Draw the circle, include the center, and draw the tangent.

Call the center C, the point of tangency of the circle and the tangent T, and call the point on the tangent P.

The points C, T, and P create a right triangle with a right angle at T.

TC = 4 and PT = 10. Use the Pythagorean Theorem to find the length PC:

     PC²  =  PT² + TC²   --->   PC²  =  10² + 4²   --->   PC²  =  116   --->   PC  =  √116

To find the size of ∠TPC:  If you use tangent:  tan(∠TPC)  =  4/10   --->   ∠TPC  =  21.8° (approx)

Let  D  stand for the cost of a hot dog and  H  stand for the cost of a hamburger.

7D + 2H  =  14.50     --->   multiply by 2   --->   14D + 4H  =     29.00

2D + 7H  =  17.00    --->   multipoly by -7  -->  -14D - 49H  =  -119.00

                                       add down:                        - 45H  =  -90

                                                                                   H  =  2.00

Substituting into the first equation:  7D + 2(2.00)  =  14.50

                                                        7D + 4.00  =  14.50

                                                                   7D  =  10.50

                                                                     D  =  1.50

A formula that works for problems like this is:

(Percent)·(Amount)  +  (Percent)·(Amount)  =  (Final Percent)·(Final Amount)

Since the final amount is 54L, the amounts that are combined to make this are:  x  and  54 - x.

Let the amount of pure acid to be x and the amount of the 10% acid solution to be 54 - x.

Pure acid is 100% acid.

(Pure acid) + (10% Solution)  =  (20% solution)

(1.00)(x) + (0.10)(54 - x)  =  (0.20)(54)

        x       +  5.4 - 0.1x  =  10.8

                   0.9x + 5.4  =  10.8

                             0.9x  =  5.4

--->  x = 6 L of pure acid to be added to 48 L of a 10% solution.

Go with Anonymous's answer because 

not only is 7 an answer because (7)(7) = 49

but also -7 is an answer because (-7)(-7) = 49      

         (a negative number times a negative number has a positive anwer).

4.64 becomes 4.64 x 10⁰  because you do not have to move the decimal point.

[ x² / (yz⁵) ]^(1/3) · z  

To rationalize the denominator under the radical sign, multiply the numerator and the denominator by y²z:

[ x²·y²z / (yz⁵·y²z) ]^(1/3) · z  

=  [ x²·y²z / (y³z⁶) ]^(1/3) · z 

=  (x²·y²z)^1/3 / (yz²)  · z 

=  (x²·y²z)^1/3 / (yz)

Set these equations equal to each other:

Since y = y   --->   (2/3)x + 3  =  (1/3)x + 6

Multiply both sides by 3:  2x + 9  =  x + 18         (to get rid of fractions)

--->                                   x + 9  =  18

--->                                         x  =  9

When x = 9:  y = (1/3)x + 6  =  (1/3)(9) + 6  =  3 + 6  =  9

They intersect in the point  (9, 9)