hectictar

ax + (a + 1)y  =  a + 2

We are given that  (4, -8)  is a solution to this equation.

That means if we substitute  4  in for  x  and  -8  in for  y ,  we will make a true equation.

So the following equation must be true:

a(4) + (a + 1)(-8)  =  a + 2

                                            We can distribute  -8  to the terms in parenthesees and simplify the left side of the equation.

4a - 8a - 8  =  a + 2

                                            Add  8  to both sides of the equation.

4a - 8a  =  a + 2 + 8

                                            Subtract  a  from both sides of the equation.

4a - 8a - a  =   2 + 8

                                            Combine like terms.

-5a  =  10

                                            Divide both sides of the equation by  -5

a  =  -2

average rate of change over the interval  [9, 10]   =   \(\frac{f(10)-f(9)}{10-9}\ =\ \frac{11014-4052}{1}\ =\ \frac{6962}{1}\ =\ 6962\)

average rate of change over the interval  [5,  8]   =  \(\frac{f(8)-f(5)}{8-5}\ =\ \frac{1491-75}{8-5}\ =\ \frac{1416}{3}\ =\ 472\)

How much greater is the average rate of change over the interval [9, 10] than the interval [5, 8]  ?

In other words...

How much greater is  6962  than  472  ?

6490  seems to be correct 

f(n)   =   3f(n - 2)  -  2f(n - 1)        If we plug in  5  for n  we get...

f(5)   =   3f(5 - 2)  -  2f(5 - 1)   =   3f(3)  -  2f(4)

Now lets work on finding  f(3)  and  f(4)

We're given:

f(1)   =   -1

f(2)   =   3

To find  f(3) ,  plug in  3  for  n  into the function:

f(n)   =   3f(n - 2)  -  2f(n - 1)

f(3)   =   3f(3 - 2)  -  2f(3 - 1)

f(3)   =   3f( 1 )  -  2f( 2 )            We know   f(1)  =  -1   and   f(2)  =  3   so we can say...

f(3)   =   3( -1 )  -  2( 3 )

f(3)   =   -9

To find  f(4) ,  plug in  4  for  n  into the function:

f(n)   =   3f(n - 2)  -  2f(n - 1)

f(4)   =   3f(4 - 2)  -  2f(4 - 1)

f(4)   =   3f( 2 )  -  2f( 3 )            We know   f(2)  =  3   and   f(3)  =  -9   so we can say...

f(4)   =   3(  3 )  -  2( -9 )

f(4)   =   27

Now that we know   f(3)  =  -9   and   f(4)  =  27   we can find   f(5):

f(n)   =   3f(n - 2)  -  2f(n - 1)

f(5)   =   3f(5 - 2)  -  2f(5 - 1)

f(5)   =   3f( 3 )  -  2f( 4 )            We know   f(3)  =  -9   and   f(4)  =  27   so we can say...

f(5)   =   3( -9 )  -  2( 27 )

f(5)   =   -81

sum of interior angles of polygon   =   180°  *  (the number of sides  -  2)

We know the sum of the interior angles  =  9000°

And let the number of sides  =  n

9000°   =   180°  *  (n - 2)

                                               Divide both sides of the equation by  180°

9000° / 180°   =   n - 2

50   =   n - 2

                         Add  2  to both sides of the equation.

52   =   n

n   =   52

the number of sides   =   52

Hi Melody!

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Thank you, Omi and heureka!

Okay, I think I see where the  (-2, -2, -1)  came from, but I am still confused about this question and I want to see what others have to say.

If it is 2D....let's label the figure like this:

△ABE  is an equilateral triangle, so every interior angle is 60°, and so

m∠EAB  =  60°

m∠ABE  =  60°

We are given that

m∠EBC  =  90°

Now we can determine that

m∠ABC  =  60° + 90°

m∠ABC  =  150°

△ABC is an isosceles triangle. so base angles are congruent, and so

m∠BCA  =  m∠BAC

The sum of the measures of the interior angles in a triangle is 180°, so

m∠BAC  +  m∠BCA  +  m∠ABC  =  180°

                                                                     Substitute  m∠BAC  in for  m∠BCA  and  150°  in for  m∠ABC

m∠BAC  +  m∠BAC  +     150°    =   180°

                                                                     Combine like terms

2(m∠BAC)  +  150°  =  180°

                                               Subtract  150°  from both sides of the equation.

2(m∠BAC)  =  30°

                                               Divide both sides of the equation by  2

m∠BAC  =  15°

In the same way, we can determine that

m∠EAD  =  15°

Then

m∠DAC   =   m∠EAB  - m∠EAD - m∠BAC   =   60° - 15° - 15°   =   30°

Happy Birthday!!!