CPhill

I find one

[ x, y]  =  [ 9, 12 ]

x^3 + 5x^2 - 9x - 45       factor by grouping

x^2 ( x + 5)  - 9(x + 5)

(x + 5) ( x^2 - 9)

(x + 5) ( x + 3) ( x - 3)

sqrt ( p + a)  + r  = k       subtract r from both sides

sqrt ( p + a)  =  k  - r      square both sides

p + a  =  ( k - r)^2

p + a  =  k^2 - 2kr + r^2      subtract p from both sides

a  =  k^2  - 2kr   + r^2  - p

2/3  <   4/5 <  6/7

Let

Chickens  = x

Pigs  = y

Sheep = z

Cows =  w

So

4x = 1y    ⇒  y = 4x 

3y = 2z   ⇒  3(4x)  = 2z ⇒   12x  = 2z  ⇒  6x = z

5z = 2w   ⇒  5(6x)  = 2w  ⇒   30x  = 2w ⇒  15x = 1w

So

1w / 15x   =     ?w / 20x

1 / 15  =  ? / 20

20 / 15  =   ?   =  4/3 cows for 20 chickens

(2) The perpendicular bisector of the line segment AB is the line that passes through the midpoint of AB and is perpendicular to AB.

Find the equation of the perpendicular bisector of the line segment joining the points (1,2) and (7,4) Enter your answer in the form "y - mx + b."

The perpendicular line will pass through the midpoint  of  (1, 2)  and ( 7,4)

The midpoint is  [   ( 1 + 7)/ 2, (2 + 4) / 2 ]   =  [ 8/2, 6/2 ]  =  ( 4, 3)

The slope between  (1/2) and ( 7, 4)  is   [  4 - 2 ] / [ 7 - 1 ]  =  2 / 6   =  1 / 3

So....the slope of the pepenicular bisector  is   - 3

And the equation of the perpendicular bisector is

y  = - 3 ( x - 4) + 3

y =  -3x + 12 + 3

y = -3x + 15 

(1)Let line L1 be the graph of 5x + 8y = -9 . Line L2  is perpendicular to line L1 and passes through the point (10,10). If line L2 is the graph of the equation y = mx + b, then find m + b.

The slope of the line in the form Ax + By  = C   is given by -A / B

So....the line 5x + 8y  = - 9   has the slope  -5/8

A perpendicular line to this one  will have a negative reciprocal slope  =  8/5

So......the equation for L2  with slope 8/5  = m  passing through (10, 10) = ( x1, y1) is

y   = m (x  - x1)  +  y1

y = (8/5) ( x - 10) + 10

y = (8/5)x - 16 + 10

y = (8/5)x  - 6

m =  8/5     and     b   =  -6

So

m +  b  =     8/5 - 6   =   8/5 - 30/5   =   - 22/5

EDIT TO CORRECT AN ERROR

For your other question, we have

y = (4/5)x - 7/10       we want to get rid of the fractions....so...multiply through by 10

10y  = 8x - 7       add 7 to both sides

7 + 10y  =  8x   subtract 0y from both sides

7 =  8x  - 10y       rearrange as

8x - 10y  =  7

In the form  Ax + By = C, the slope of the line is  -A / B

So....rearranging the given line as

2x + 3y  = 7......the slope is =  -A / B  =    -2 / 3

So....a perpendicular line will have  a negative reciprocal slope =   3 / 2  =  m

Because OA  and OC are radii of the circumcircle, then in triangle OAC,  angle OAC  = angle ACO  = 32

So....angle AOC  =    [ 180   -  2 measure of amgle OAC ] / 2  =  [ 180 - 2*32 ] / 2  =

[  180 - 64 ] / 2  =     58°