Final answer:
It is impossible for the numbers of buttons in the Blue Jar (x), Green Jar (y), and Yellow Jar (z) to have the given totals because adding these together yields an inconsistent system of linear equations, pointing to a contradiction in the given values.
Step-by-step explanation:
The question is asking to determine why it is impossible for the combination of numbers in the Blue Jar (x), Green Jar (y), and Yellow Jar (z) to coexist with the given totals for pairs of jars. The totals provided are 30 for (x + y), 8 for (y + z), and 40 for (x + z). This is a problem of system of linear equations, and we can analyze the feasibility of this system being consistent.
If we add all three given totals together, we get the sum of (2x + 2y + 2z) = 78. If we take half of this sum to correct for the doubling of each variable, we should get the total number of buttons (assuming x, y, and z are each counted twice). However, 78 divided by 2 equals 39, which must be the sum of all the buttons in the three jars. But if we subtract the number of buttons in the Green and Yellow jars (8) from this total, we would be left with 31 buttons for Blue and Green jars, which contradicts the given total of 30 buttons for these two jars. Hence, there's a contradiction, meaning the system of equations is inconsistent and it's impossible for the given numbers to be correct at the same time.