proyaop

The ratio of yellow paint to blue paint in the special green paint is 3:5. Each liter of special green paint is 3/8 yellow paint, and 5/8 blue paint. Thus if we set up our variables y and b respectively for yellow paint and blue paint, to make 'n' of Green Goober's special green paint, we will need 3ny/8 + 5nb/8 paint. 

3ny/8 + 5nb/8 = n where n is the amount of green goobers special green paint.

a) Plug in 10 into the expressions, then we can see whether or not they are equal. 

b) Plug in -3/2 and see whether or not the two expressions are equal.

c) Set the two expressions equal to each other, cancel out the denominators, combine like terms, and you will see that the two expressions equal to each other EXCEPT when x is -3/2. 

The inverse of a function would be reflecting it over the line y = x. Thus, we can substitute, so we get x = y^2 - 4 as the inverse function.

Given that equation, you can find the values of x and the function of y. :)

2x = 5 * 0.08206 * 300

2x = 5 * 8.206 * 3

x = 5 * 3 * 4.103

x = 20.515 * 3

x = 61.545

How many ways can I put 4 identical balls into boxes A, B, and C?

This is the same as saying how can I distribute 4 pieces of candies to Alex, Bob, and Claire. Since there are no constraints of "at least" how many balls can go to each box, we can use STARS AND BARS to solve this.

A + B + C = 4 and A, B, and C are non-negative.

(4 + 3 - 1) choose (4 - 1) = 6 choose 3 = 20.

There are 20 different ways to put 4 identical balls into 3 distinguishable boxes.

f(x) = sqrt(x)

f(x/2) = stretch horizontally by factor of 2.

f(x/2) + 3 = stretched horizontally by factor of 2 and y-intercept raised by 3 units.

You apply it... (equation is bolded above) :)

Isolate y:

y = -5x + 31

If you want to solve the entire system of equations, multiple E1 by 3, and E2 by 5

15x + 3y = 93

-15x + 20y = -185

Adding both equations I obtain:

23y = -92

y = -4

x = 7

(x, y) = (7, -4)

Draw a table: (or solve algebraically)

Week 5: 2 wins 13 losses

Week 6: 4 wins 14 losses

Week 7: 6 wins 15 losses

Week 8: 8 wins 16 losses

Week 9: 10 wins 17 losses

Week 10: 12 wins 18 losses

...

Week 13: 18 wins 21 losses

Week 14: 20 wins 22 losses

Week 15: 22 wins 23 losses

Week 16: 24 wins 24 losses

To solve algebraically:

f(n) = 2n - 8 wins, n + 8 losses 

2n - 8 = n + 8

n = 16

2 + 1/6 = 12/6 + 1/6 = 13/6

1 + 2/3 = 3/3 + 2/3 = 5/3

1 + 1/6 = 6/6 + 1/6 = 7/6

You could use mass points and do a lot of shenanigans, or you could use a little bit of an intuitive guess. 

Upon seeing the numbers 6 and 8, I assumed that angle A was 90 degrees. 

It works out pretty well, but that's the hint! Good luck mrinank...