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Yeah exactly

3x +  y  =   13

x +  6y   =  -7

A)    Multiply  the  second equation  through by   -3

3x   +  y   =   13

-3x  - 18y  =  21

B)  Add the two equations and we get that

        -17y   = 34

             y  = 34  / -17   =   -2 

C  and D  )    x  +  6(-2)   =  -7

         x  -   12  =    -7

          x  =  -7  + 12   =  5

E )  (x , y)   = ( 5, -2)

You have 9    B has 6

Tan^-1 (3/5)=

atan(3/5)  = 30.963756532074 degrees

y  =x +  7

x + 3y  =  9

A)   x +  3 ( x +7)  =  9

B )    x + 3x +  21   = 9

          4x   +  21    =  9

           4x   =    9  - 21

           4x  =  -12

             x =  -12  /  4   =   -3

C )    y  = -3 + 7  =  4

D )   ( x , y)    =   (  -3 , 4)

There are many methods to solve this problem, but the method I'm using today is the one shown below. As you can see, drawing a rectangle around the perimeter of the triangle highlights 3 smaller triangles that we can use to help solve the problem. First, we find the area of the rectangle, which is 4x6=24 sq units. Next, we find the area of triangles A, B, and C, which are 3x2/2=3 sq units, 3x4/2=6 sq units, and 6x2/2=6 sq units, respectively. Now we can subtract the areas of those triangles from the area of the rectangle, 24-6-6-3=9 sq units, giving us the final answer: 9 sq units. Hopefully, this helps! - BasicMaths

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." - William Paul Thurston

CE = 2*3/4 = 6/4 = 3/2.

AB XX

BA XX

X AB X

X BA X

XX AB

XX BA      SIX ways to be next to each other   *  2 ways to fill XX =   12 ways

80-25= 55 remaing

triangle side rule       sum of any two sides must be greater than third side

39 and 16      25+16> 39

16 would be smallest whole number side length  

Okay, so this is a problem involving systems of equations. We would set up this system as follows:

n+d=21 (number of coins)

5n+10d=150 (amount of money)

For the second equation, I find it easier to remove the decimal places altogether and write it in cents instead. We can then rearrange the first equation as n=21-d and substitute that into the second. The second equation then becomes 5(21-d)+10d=150. Writing the terms out and simplifying gives us 5d=45, which becomes d=9. Therefore, we have 9 dimes. Plugging that value into the first equation gives us n=12. This means we have 12 nickels. We then double-check, and it all works out. Hopefully, this helps! - BasicMaths

(note: in the system of equations above, n stands for nickels and d for dimes)

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." - William Paul Thurston

-16t^2  +  96t  +  880   =  0        multiply through  by   -1

16t^2  -  96t  -  880   =  0       divide through  by  16

t^2  - 4t  -   55  =  0                 factor

(t  - 11)  ( t + 5)   =  0

The first factor set to   0   and solved for t   will  give us  the answer

t  - 11   = 0

t  =  11   seconds

P(x) = -54x^2 +162x -108

(a)  The  x  that gives  the max profit  is     -162 / (2 * -54) =  -162  /  -108   =  1.5 =  1500  units

(b)  Max profit is  -54(1.5)^2  + 162 ( 1.5)  -  108    =  13.5   =   $13,500

(c)  See the graph  here :  https://www.desmos.com/calculator/lsabrvvq2g

The  least  number of units  that  should  be produced  is   at  (1,0)  =  1000 units

The  greatest number  of units  that should be produced is at ( 2,0)  =  2000 units

Two

CADB   and   ADBC

z   =  -3              multiply both sides by 6

6

z = -18

Use quadratic formula to find https://web2.0calc.com/questions/power-meassurement

the zeroes of the equation...between these zeroes.. this is 

the portion of the equation above zero....where profit is made ( positive)

5 4/5 =   29 / 5        Inverse is   5/29

When rocket hits ground , height =0

0 = -16 t² + 96t + 880    Quadratic formula to find t....   throw out negative t value

Max occurs at x = -b/2a = - 162 /(2 * -54)= 1.5  hundred units  = 1500 units

use this value of x =1.5 in the equation to calculate the profit (in thousands of dollars)

P(x) = -54 (x^2 - 3x + 2)

P(x) = -54 (x^2 - 3x + (3/2)^2 + 2 - 9/4)

P(x) = -54(x - 3/2)^2 + 27/2

Number of units = 1.5 hundred

Maximum profit = 13.5 thousand dollars

Try to do part (c) on your own.

499 acres = 2.02 km²          39000/2.02 = pop density of Monaco in ppl/km²

                                                   1000/ .44 = pop density of Vaticanin ppl/ km²   
you should finish from here .....

\(x^2 + y^2 = 2x - 6y + 6 +4x - 8y\\ x^2 + y^2 = 6x - 14y + 6\\ x^2 - 6x + y^2 + 14y - 6 = 0\\ (x^2 -6x + 9) + (y^2 +14y + 49) = 64\\ (x - 3)^2 + (y + 7)^2 = 8^2 \)

The graph is a circle centred at (3, -7) and with radius 8 units.

The required area is \(\pi \cdot 8^2 = \boxed{64\pi\text{ sq. units}}\)

"In a set of four numbers, the first three are in a geometric sequence and the last three are in an arithmetic sequence with a common difference of 6. If the first number is the same as the fourth, find the four numbers."

I have this question too! :( I thought there would be a answer.

Draw  MJ  perpendicular  to  PR  to meet PR  at J

Draw  MK  parallel  to  PR to meet QN  at  K

MK  divides  QN  into   equal  parts  so   KN  =  6 =   MJ  

And

PJ  / MJ  = PN /NQ

PJ/ 6  = 9/12 =  3/4

PJ  =  18/4  =  9/2  = 4.5

So  RJ   =  18 - 4.5      = 13.5

And

ON / NR   = MJ / RJ

ON / 9  = 6 / 13.5

ON  =  54/13.5   = 4

So using the Pythagorean Theorem:

OR  = sqrt  ( NR^2  +   ON^2)  =  sqrt ( 4^2  + 9^2)    = sqrt (97)

5/3  =  1-2/3  

2π   =  6.28  

The whole numbers between them are 2,3,4,5,& 6 and  

the question is "how many are there" so the answer is 5  

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