CPhill

6mn  = 27 - 3m - 2n     add 2n to both sides

6mn + 2n  =  27 - 3m      factor this

2n ( 3 + m)  =  3(9 - m)

Because of "2n"....the left side is always even

And the right side will be even when "m" is odd

Divide both sides by 3

(2/3)n (3 + m) = 9 - m

The right side is an integer

The left side will be an integer when "n" is a multiple of 3

Note that when n  = 3 and m  = 1, we have that

(2/3)(3)(3 + 1)  =  9 - 1

2(3 + 1)  =  9 - 1

2(4)  =  8       which is true

So....(m, n)  =  (1, 3) 

We have 20 ways to pick a president, 19 to chose a vice-president and 18 ways to choose a treasurer....so

20 * 19 * 18   =   6840 ways

We have   4 possibilities for the hundreds digit [6, 7, 8 or 9], 3 posibilities for the  tens digit [ 5, 6 , 7 ]  and the units digit is 2  [ the smallest prime]

So....the number of 3-digit numbers satisfying these conditions is :

4 * 3 * 1   =   12  possibilities

The A can appear in any one of 3 positions when it is used once

And there are 25 * 25  posiibilities  for the other two positions

So..... C( 3, 1) * 25^2    = 1875      

When the word has 2 A's....they can appear in any two of the three positions

And the the other position can be filled in 25 ways

So  C(3,2) * 25  =  75    

And  only one word has three A's  

So...the total possible words  =   1875  + 75  + 1  =    1951

We have 2 choices for the first, 4 for the second and 2 for the third....so...

2 * 4 * 2   =     16 different combos

The slope of the line  is    [ 11 - 7 ] / [ 3 - 1]  =   4/2  =  2

So....using the point (1,7) we can write the equation of a line passing through the two given points as

y = 2(x - 1) + 7

y = 2x - 2 + 7

y  = 2x + 5

In the form y  = mx + b,  "b"  is  y intercept....and where the line intercepts the y axis, the x value of the coordinate  = 0

So

(0, 5)  is the y intercept 

Assuming that   4(x - 2)  and (5x - 20)  are the side lengths.....because all sides of a square are equal, we have

4(x - 2) = 5x  - 20

4x  - 8  =  5x  - 20        add 20 to both sides, subtract 4x from both sides

12  =  x

So....the side length  is   4(12 - 2)  =  4 * 10   =  40 cm

4x - y + 4  = 0

The x intercept  wil occur when y  = 0.....so....

4x - 0 + 4  = 0

4x  + 4  =   0       subtract 4 from both sides

4x  =  - 4           divide both sides by 4

x  = - 1

The x intercept for f(x)  is  at x = -3

So..... 4x - y + 4  = 0    has the larger x intercept at  x  = -1   ⇒   B

No quadratic graph appears

Slope  =   [ subtract the y coordinates in any order ]

               ___________________________________

                [ subtract the x coordinates in the same order]

 [ 2  (-2) ]                4

_______   =       _______

[- 4 - (-7)]                3

To find the y value that makes the second segment parallel to the first, we have

[ 4 - y ]              4

______   =      ___                cross multiply        

[ 7 - 4 ]             3

3[ 4 - y ] = 4 [ 7 - 4 ]

12 - 3y  =  12           subtract 12 from both sides

-3y  =  0         divide both sides by  -3

y  = 0