First, let's call the number of blue marbles Rosy had first bb and the number of white marbles Rosy had first ww.
At first Rosy had:
b = 4wb=4w - she had 4 times as many blue marbles as white.
Then Rosy had:
b - 18 = w + 6b−18=w+6 - she gave away 16 blue marbles and got 6 white marbles she had the same number blue and white marbles.
We can substitute 4w4w from the first equation for bb in the second equation and solve for ww to find the number of white marbles Rosy had at first:
b - 18 = w + 6b−18=w+6 becomes:
4w - 18 = w + 64w−18=w+6
4w - 18 + color(red)(18) - color(blue)(w) = w + 6 + color(red)(18) - color(blue)(w)4w−18+18−w=w+6+18−w
4w - color(blue)(w) - 18 + color(red)(18) = w - color(blue)(w) + 6 + color(red)(18)4w−w−18+18=w−w+6+18
4w - 1color(blue)(w) - 0 = 0 + 244w−1w−0=0+24
(4 - 1)color(blue)(w) = 24(4−1)w=24
3w = 243w=24
(3w)/color(red)(3) = 24/color(red)(3)3w3=243
(color(red)(cancel(color(black)(3)))w)/cancel(color(red)(3)) = 8
w = 8
Now, we can substitute 8 for w in the first equation to calculate the number of blue marbles Rosy had at first.
b = 4w becomes:
b = (4 * 8)
b = 32
At first Rosy had 32 blue marbles and 8 white marbles.
When she gave away 18 blue marbles away and received 6 more white marbles she had 14 blue marbles and 14 white marbles.
