Melody

ok, your first step is fine

Solve the system of equations

5x-2y=16

15x-7y=51

Enter answer as ordered pair (x,y).

This is what I did:

-3(5x-2y=16)

-15x+6y=-48

15x-7y=51

so far so good  :)

   I am going to sid track for a bit

        You know that   24 = 2 dozen

        and you know that    7 = 21/3

        these are completely unrelated facts but from this knowledge I can say   24+7 = 2 dozen + 21/3

        Think very hard about what I have done.  I added the left sides and that will = the sum of the right side.

So

you have

-15x+6y=-48

15x-7y=51

these are 2 independent facts

Add the left sides and that will equal the sum of the right sides

-15x+15x+6y+-7y = -48+51

-1y=3

y=-3

Now you know that y=-3 just sub it into one of the original equations to get x

5x-2y=16

5x-2*-3=16

5x+6=16

5x=10

x=2

so I have x=2 and y = -3

I have substituted them into both original equations to check and they work just fine.

Your second question you have asked is almost identical.   Just easier.

I already answered this quite recently.

Anyway, you only have to order the 3 species so that is 3!=6

You are welcome :)

This was quite fun to solve.

I am not going to do the whole thing for you but I will give you a good outline.    

I put a letter in each of the boxes

a,b,c    in the top row

d,e,f     in the second row

g,h,i     in the bottom row.

Now I looked just at the solutions for a

a=1/3(a+b+c)       and     a=3/4(a+d+g)

by simplifying these and looking at possible solutions

I determined the value for a,  d and g

and that b+c =12   which meant that both b and c had to be between 3 and 9. And c=12-b

I redrew the boxes with all the info that I just gleaned.

Next I looked at the two equations for e  and from those I get the value of e and f and I found that h=4-b

Last  I looked at the equations for i and finished the problem.

Finally I checked my answers and they all worked.

Here is the answer but I don't know how to do it.

a and b are unknown.

you keep saying a=10 and b=10 but that is only one of an infinite number of possibilities.

a could be 8 and b 12

or a could be 5.3 and b could be 14.7

or a could be pi and b could be 20-pi

What you know is that a+b=20    which means   b=20-a

You want the sum of their squares is  a^2 +b^2   but we have to replace b with  20-a

so

\(a^2+b^2\\ =a^2+(20-a)^2\\ =a^2+400+a^2-40a\\ =2a^2-40a+400\)

The equation of the line AB is     \(y=\frac{-31}{32}x+31\)

The equation of the circle is     \(x^2+y^2=23^2\)

I want to find the intersection.

So I need to solve

\(x^2+(\frac{-31}{32}x+31)^2=23^2\)

You can solve this as a standard quadratic if you want to but it is pretty awful so I used the wolframalpha calculator and go the the approx answers of     x=11.351 and x=19.635

So between these values of x, the signal will be available    19.635-11.351 = 8.284

8.284/32 = 0.2588

So the signal will be available for approximately  25.88% of the drive.

calculation:

b=20-a

"Write an expression for the sum of their squares."

a^2+b^2

=

now do the substitution and then simplify.

I will start you off.

a+b=20

b=20-a

Which bit do you need help with?

Show us what you have tried for yourself.