[vendors] sell an herbal tea blend. By weight, Type I herbal tea is 30% peppermint, 40% rose hips and 30% chamomile, Type II has percents 40%, 20% and 40%, respectively, and Type III has percents 35%, 30% and 35%, respectively. How much of each Type of tea is needed to make 2 pounds of a new blend of tea that is equal parts peppermint, rose hips and chamomile?

I set it up like this: I: x, II: y, III: z

.3x + .4y + 3.5z = 2/3

.4x + .2y + .3z = 2/3

.3x + .4y + 3.5z = 2/3

The way I set it up is that E1 is all the peppermint, E2 is all the rose hips, and E3 is all the chamomile. The question stipulates that these must be in equal proportions in the final 2 pound mix: 3 * 2/3 = 6/3 or 2 pounds. To work the problem I multiplied each through by 100 to clear most of the fractions--leaving 200/3 for the constants. Then proceeded to put the equations in "triangular form". You are ok with E1 and E2 but at E3 you end up with 0 = 0 which tells you that Z is a free variable and you need a parametric solution as explained above. My final equations looked like this:

x + 4/3y + 7/6z = 20/9

y + 1/2z = 26/15

0 = 0

You see that E1 were originally E3 are identical, and where I'm from that leads to a "parametric solution". z is a "free variable" so the equation is "dependent". You set z equal to t, then express x and y in terms of t, then give your solution set via the "set builder" method of describing a set: (-4/45-1/2t, 26/15 - 1/2t, t | 0 < t < 4/3). The first three expressions separated by commas represent x, y and z respectively, and the second part just says "such that t is a real number between 0 and 4/3" or something like that. I actually said -infinity < t < infinity. But my answer is wrong. Book says (4/3 - 1/2t, 2/3 - 1/2t, t) for 0 < t < 4/3).

That's all I have done. Where did I go wrong?