Solving the Impossible Button Puzzle
Why is it impossible for the number of buttons in the Blue Jar(x) and Green Jar(y) to be 30, in the Green Jar and Yellow Jar(z) to be 8, and in Blue and Yellow 40?
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.
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.
Answer:
Given the totals for the combinations of buttons in the three jars, it is impossible for the numbers to add up as stated. This impossibility arises from the contradictory nature of the system of linear equations formed by the totals.
When we add the totals for the pairs of jars together, we get:
(x + y) = 30
(y + z) = 8
(x + z) = 40
If we sum up these equations, we get:
(2x + 2y + 2z) = 78
However, when we divide this sum by 2 to represent the total number of buttons in all three jars (assuming each variable is counted twice), we get:
(x + y + z) = 39
Now, if we subtract the number of buttons in the Green and Yellow jars (8) from this total, we are left with 31 buttons for the Blue and Green jars. This contradicts the given total of 30 buttons for these two jars, leading to the inconsistency in the system of equations.
Therefore, based on the mathematical analysis of the given totals, it is evident that it is impossible for the numbers of buttons in the Blue Jar, Green Jar, and Yellow Jar to align with the provided sums simultaneously.