find the number of red, blue and yellow stars that are in the bag

Question

doing a sample mystery bag assignment

these are the steps that were given to me

the sum of red and blue is yellow

blue squared is red

blue squared plus red is yellow plus twelve

yellow divided by blue is red minus eleven

blue cubed is the difference between four times yellow and red

Answer 1

R + B = Y

B^2 = R

B^2 + R = Y + 12

Y/B = R - 11

B^3 = 4Y - R

Look at these this line:

B^2 = R

In the other lines replace all Rs with B^2:

B^2 + B = Y

B^2 + B^2 = Y + 12

Y/B = B^2 - 11

B^3 = 4Y - B^2

Let's re-word the third line:

B^2 + B = Y

B^2 + B^2 = Y + 12

Y = B^3 - 11B

B^3 = 4Y - B^2

See the first line?  Let's replace all of the Ys with B^2 + B:

B^2 + B^2 = B^2 + B + 12

B^2 + B = B^3 - 11B

B^3 = 4(B^2 + B) - B^2

Simplify:

2B^2 = B^2 + B + 12

B^2 = B^3 - 12B

B^3 = 4B^2 + 4B - B^2

More simplifying:

B^2 = B + 12

B^2 = B^3 - 12B

B^3 = 3B^2 + 4B

Consider the first line:

B^2 = B + 12

Move everything to one side:

B^2 - B - 12 = 0

Same as:

(B - 4) (B + 3) = 0

B has to equal 4 or -3.

You can't have negative 3 of a thing, so B = 4

Consider the other two lines:

B^2 = B^3 - 12B

B^3 = 3B^2 + 4B

Check those to make sure B = 4 works.

4^2 = 4^3 -12*4 >> 16 = 64 - 48 >> Yes.

4^3 = 3*4^2 + 4*4 >> 64 = 48 + 16 >> Yes.

Remember how we said B^2 + B = Y ?  Now that we know B = 4, we can do this:

4^2 + 4 = Y

16 + 4 = Y

Y = 20

Remember how we said B^2 = R ?  Now that we know B = 4, we can do this:

4^2 = R

R = 16

Answer:  16 red, 4 blue, 20 yellow.