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Welp, this is an easy case of the Chicken and cats problem (is it called that? Idk)
Let's start with calculating what would happen if only 123 adults came. 123 x 7 = 861 dollars. That's only 32 more dollars than they made.

Now, children cost 2 dollars less than adults (that sounds weird, but I'm too lazy to correct it). Therefore, if we divide 32 by 2, we would get 16. 

This means there are 

16 Children

107 adults. 

We double check this: 107 x 7, 16 x 5. 80 + 749 = 829. Thats the answer. 
(I could be wrong, Idk) 

\(\frac{(\frac{1}{y} - \frac{1}{x})}{\frac{1}{y} + \frac{1}{x}}\)

\(= \frac{\frac{x - y}{xy}}{\frac{x+y}{xy}}\)

\(= \frac{x-y}{x+y}\)

I think this is right..

I do not know what the answer is, but, in listing some sets, I get more than 56 sets when counting only those that include the number 1.

A guy wire runs from the top of a cell tower to a metal stake in the ground. Malika places a 12-foot tall pole to support the guy wire. After placing the pole, Malika measures the distance from the stake to the pole to be 5 ft. She then measures the distance from the pole to the tower to be 23 ft. Find the length of the guy wire, to the nearest foot.

If                 2·sin(A)  =  5·cos(A)

then  sin(A) / cos(A)  =  5/2

                      tan(A)  =  5/2

x^2 + y^2 = 2x + 2y

x^2 + y^2 - 2x - 2y = 0

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

x = 2 and y = 2, and x = 0, y = 0 works. 

So r = 0+0 and 2+2, but it has to be positive, meaning the answer is 4. :))

=^._.^=

Thank you. :))

=^._.^=

There are 56 such sets.

Draw the line:  y  =  (3x + 20) / 5.

From the origin draw a line that passes through the origin that is perpendicular to the original line.

The point of intersection of these two lines will be the center of the chord of the circle.

The slope of  y  =  (3x + 20) / 5  is  3/5.

The slope of the line perpendicular to this line is  -5/3.

The perpendicular line passes through the origin, so its equation is  y  =  -5/3·x.

The intersection of  y  =  (3x + 20) / 5  and  y  =  -5/3·x  is the point  (-30/17, 50/17).

The distance from this point to point A is 10.

The distance from this point to the origin is  sqrt(  (-30/17 - 0)²  +  (50/17)²  )  =  sqrt( 900/289 + 2500/289 )

                     =  sqrt( 3400 / 289 )

Now, find the distance from the origin to point A by using the Pythagorean Theorem; this will be the radius of the circle.

If you let  x = 5,  then you get:  37 x 12 x 5  =  2220

There are a total of 24 nuts.

The number of ways to draw 6 from 24 is:  ₂₄C₆.

There are 5 walnuts.

The number of ways to draw 3 from 5 is:  ₅C₃.

There are 19 nuts which are not walnuts.

The number of ways to draw 3 from 19 is:  ₁₉C₃.

You need to select 3 walnuts and 19 nuts which are not walnuts from a total of 24 nuts.

The probability of doing this is:  ₅C₃ · ₁₉C₃  /  ₂₄C₆.

Your answers will be the values on the vertical (attendance) axis.

For example, for 2009, the attendance was 27; for 2020, the attendance was 5; etc.

x + y  =  10     --->    (x + y)2  =  (10)2     --->     x2 + 2xy + y2  =  100

Combining this with  x2 + y2  =  56     --->     x2 + 2xy + y2  =  100

                                                                      x2           + y2  =    56

    subtracting:                                                               2xy  =  44

                                                                                        xy  =  22

Since  x + y  =  10     --->     y  =  10 - x.

Combining these two equations:  xy  =  22     --->     x(10 - x)  =  22

                                                                                  10x - x²  =  22

                                                                                x² - 10x + 22  =  0

Using the quadratic equation:  x  =  5 + sqrt(3)     --->     y  =  10 - ( 5 + sqrt(3) )  =  5 - sqrt(3)

Also,  x  =  5 - sqrt(3)     and    y  = 5 + sqrt(3)

In a right triangle    for the two non-right angles       sin b = cos c = 4/17

thank you!

Since the area of the square is 100, each side must equal 10.  [10² = 100]

Since each side of the square is 10, the diameter of the circle must be 10, so the radius of the circle must be 5.

This means that the area of the circle is:  Area  =  pi · radius²  =  pi · 5²  =  25·pi

The shaded region is the area of the whole square minus the area of the circle:  100 - 25·pi

Since  f(x)  =  x² - x - 3     --->     f(-2)  =  (-2)² - (-2) - 3  =  3.

Since t(x)  =  9 + 2·f(x)   --->     t( f(-2) )  = 9 + 2·(3)  =  15

btw i accidentally left this part out its "to enter the solutions (2,4) and (-3,9), you-"

Since  cos B  = b / a  and  cos B  =  3 / 25,  b  =  3  and  a  = 25  (or some multiple).

Since  c² + b²  =  a²,  c² + 3²  =  25²     --->     c² + 9  =  625     --->     c²  =  616     --->     c  =  sqrt(616).

Since  cos C  =  c / a,  cos(C)  =  sqrt(616) / 25

that seems to be incorrect

Solving the equations, we get the solutions are (5 + sqrt(7), 5 - sqrt(7)), (5 - sqrt(7), 5 + sqrt(7)).

Nice work, catmg !   

Here's a hint:

sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

cos^2(x)+sin^2(x) = 1

tan(x) = sin(x)/cos(x)

=^._.^=