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You too!

No problem! Have a good day!

Appreciate your help MS

We don't actually want to substitute the $y=31-5x$ into the second equation I think you meant substitute into $y$.  After this we get

$-3x+4(31-5x)=-37$ for the second equation. Distribute to get

$-3x+124-20x=-37$ Combine like terms to get

$-23x+124=-37$ Then we subtract $124$ from both sides resulting in

$-23x=-161$ Divide by $-23$ from both sides to get

$x=7$ Plug this into the first equation resulting in

$5(7)+y=31$ Distribute to get

$35+y=31$ Subtract $35$ from both sides to get

$y=-4$. So our answer is 

$(7,-4)$

By the Binomial Theorem, the constant term is 220.

Yes, why?

I have looked at Tiggsy's solution and I have a query.

If anyone knowledgeable wants to answer me that would be good.

This site looks very relevant but I have not managed to work through it properly yet.

Hi Tiggsy,

This question has been referred to recently and I have tried to understand you logic.

This statement confuses  me.      and is 2/3 of the way down that line from P.

Where did the 2/3 come from?   Is this just some fact that I am supposed to know?

Tiggsy is not here a lot so if someone else can explain this I would appreciate it.

(Please do not respond if you do not know what you are talking about)

Thanks Ginger,

NOW our answers almost match.

So there are 2304 possible outcomes (for 2 consecutive throws) but we only care about the ones inside this Venn diagram

So there are 36 ways to get six heads 

and 64 ways to get two 6s.

and one way to get 2 sixes AND six heads.

So there is 35 ways out of  99 relevant outcomes where 6 heads is thrown BEFORE 2 sixes.

P(HHH,HHH before 6,6) =   \(\frac{35}{99} = 0.\bar{35}\)

The logic I used last time was more complicated but similar. I think I just made careless errors.

On the graph of y = (x + 2)^4 - 500, how many points are there whose coordinates are both negative integers?

Hello Guest!

Two
namely:  \(P_1(-1,-419)\\ P_2(-2,-244)\)

  !

Also looked at here :

Write an equation of the line passing through the point(2,10)
that is perpendicular to the line 2x+y=17​

Hello Guest!

\(\color{BrickRed}f_1(x)=y=-2x+17\\ m_1=-2\)

\(m_2=-\frac{1}{m_1}\\ \color{blue}m_2=\frac{1}{2}\)

\({\color{blue}f_2(x)=y=m_2(x-x_1)+y_1}=\frac{1}{2}(x-2)+10\)

\(y=\frac{1}{2}x+9\)

  !

1 week x 7 days x 24 hours x 60 minutes x 60 seconds =604,800 seconds in 1 week.

Orbital Speed of Earth =~30 km/second =~18.6 miles / second

604,800 x 18.6 =~11,249,280 miles in 1 week.

Sides: a = 15   b = 3.606   c = 15.62

Area: T = 27
Perimeter: p = 34.226
Semiperimeter: s = 17.113

Angle ∠ A = α = 73.503° = 73°30'12″ = 1.283 rad
Angle ∠ B = β = 13.325° = 13°19'30″ = 0.233 rad
Angle ∠ C = γ = 93.172° = 93°10'18″ = 1.626 rad

Height: ha = 3.6
Height: hb = 14.977
Height: hc = 3.457

Median: ma = 8.5
Median: mb = 15.207
Median: mc = 7.616

Call the apex angle  A

The  left base angle B

The right base angle C

Let the altitude meeting base (3 + 5)  =  8  be DC

Let the altitude meeting base 2 + x  be EB

Note  that right   triangles  AEB   and   ADC are similar

This implies that    AB/ AE  =  AC/ AD 

So

8 / 2 = (2 + x) /  5      

4  = (2 + x)  / 5

5*4  = 2 + x

x = 5*4  - 2

x   = 18    ( just as  jugoslav found  !!! )

Good job, imnotamaster   !!!!

The hypotenuse  = 2 * ( side opp the 30°  angle )  = 2 (6)  = 12

Area of equilateral triangle =  (1/2) (12)^2 sqrt (3)  / 2 =   36sqrt 3  units^2

OK man.....glad to help   !!!

Area in northwest quarter  =  880 / 4   = 220 acres

(1/2) of this  =110 acres

Acres left = 880  -  110   =   770

Let   A   =(0,5)    F  ( 3, 0)

Slope of AB =  -5/3

Equation of line through  this  segment =  y = -(5/3)x  + 5

Let E = ( 0,3)      B = ( 7,0)

Slope of BE =  -3/7

Equation of line through BE

y = (-3/7)x + 3

Find D  = the intersection of these two lines

(-5/3)x  + 5 = (-3/7)x + 3

5 -3 = ( 5/3 - 3/7)x

2  = (26/21)x

x = 21/13

y= (-3/7)(21/13) + 3

y = 30/13

Slope of AB =  -5/7

Equation of line through AB

y= (-5/7)x + 5   Put into standard form

7y= -5x + 35

5x + 7y  - 35  =  0

Distance  from D to AB  is  given by

l 5 (21/13)  + 7 (30/13)  -35  l               140/13          140 / [ 13sqrt (74) ]  =  altitude of ADB

_________________________   =    _________  =  

      sqrt ( 5^2 + 7^2)                             sqrt ( 74)

AB length =  sqrt  [ 5^2 + 7^2] = sqrt (74)

Area of  ADB  = (1/2) AB * altitude  =  

(1/2) sqrt (74)  *  140 / ( 13 sqrt (74) )  =    140/26  =  70/13

Number of identifiable  "words"  without the T   = 

       10!

_____________  =    75600

3!  * 2!  * 2!  * 2!

Since we can append the T  at either the first or last of each of these....

75600  *  2    = 

151,200  "words"

The vertex  is  (-1, -1)

So we  have the form

4p ( y + 1)  = (x + 1)^2

And the point (1,3)  is on the  graph so 

4p ( 3 + 1)  = ( 1 + 1)^2

16p  = 4

p =  4/16 = 1/4

The equation  is

y  + 1 = ( x + 1)^2

y = x^2 + 2x + 1 - 1

y = x^2 + 2x

A = 1   B   = 2   C   = 0 

A + B + C =    3

Ok well good job then!