If @ + * = 16 + # – & = 12 @ – & = 16 then which one of the following

The second equation `+ # – & = 12` appears to have a leading ‘+’ which is likely a typo. Given the context of the other two equations, the most probable intended form for the second equation is either `# – & = 12` or `* + # – & = 12`.

Let’s analyze the system assuming the second equation is `# – & = 12`:
1) @ + * = 16
2) # – & = 12
3) @ – & = 16

From equation (3), we can express @ in terms of &: @ = 16 + &.
From equation (2), we can express # in terms of &: # = 12 + &.
Substitute @ = 16 + & into equation (1):
(16 + &) + * = 16
16 + & + * = 16
& + * = 0
This implies * = -&.

The relationships between the variables are:
@ = 16 + &
* = -&
# = 12 + &

This system has one degree of freedom, parameterized by &. There are infinitely many solutions unless there are unstated constraints. Let’s check which option leads to a valid solution within this framework:

A) @ = 9:
If @ = 9, then from @ = 16 + &, we get 9 = 16 + & => & = 9 – 16 = -7.
Now find * and # using & = -7:
* = -& = -(-7) = 7.
# = 12 + & = 12 + (-7) = 5.
Let’s check if the values (@=9, *=7, #=5, &=-7) satisfy the original equations:
1) @ + * = 9 + 7 = 16 (Correct)
2) # – & = 5 – (-7) = 5 + 7 = 12 (Correct, assuming this form of eq 2)
3) @ – & = 9 – (-7) = 9 + 7 = 16 (Correct)
So, the solution (@=9, *=7, #=5, &=-7) is valid if the second equation is `# – & = 12`. In this solution, the statement @=9 is true.

Let’s check other options against the relationships @ = 16 + &, * = -&, # = 12 + &:
B) * = 7:
If * = 7, then from * = -&, we get 7 = -& => & = -7.
This gives the same value for & as option A, leading to the same solution (@=9, *=7, #=5, &=-7). In this solution, * = 7 is also true.

Since both A and B are true in the same valid solution set under this interpretation, there is likely an issue with the question, as only one option should be correct.

However, if we assume the intended correct answer is A, it implies that the system must have a solution where @=9, and this solution is the one the question is probing. Given the ambiguity, and that option A provides a specific value for one variable which leads to a consistent (though not unique system-wise) solution, we proceed with the derivation showing that @=9 is possible. The calculation above shows that if @=9, then &=-7, *=7, and #=5, which satisfies the system (with the second equation interpreted as `# – & = 12`).

Let’s also consider the alternative interpretation where the second equation is `* + # – & = 12`:
1) @ + * = 16
2) * + # – & = 12
3) @ – & = 16
From (1) and (3), subtracting (3) from (1) gives: (@ + *) – (@ – &) = 16 – 16 => * + & = 0 => * = -&.
Substitute * = -& into (1): @ + (-&) = 16 => @ – & = 16, which is (3).
Substitute * = -& into (2): (-&) + # – & = 12 => # – 2& = 12 => # = 12 + 2&.
Relationships: @ = 16 + &, * = -&, # = 12 + 2&.
A) @ = 9: 9 = 16 + & => & = -7. (@=9, *=7, #=12 + 2(-7) = 12 – 14 = -2, &=-7). Solution (@=9, *=7, #=-2, &=-7). This solution satisfies all three equations, assuming the form `* + # – & = 12` for the second one. In this solution, @=9 is true.
B) * = 7: 7 = -& => & = -7. (@=9, *=7, #=-2, &=-7). * = 7 is also true in this solution.

Again, options A and B are simultaneously true. Assuming option A is the intended correct answer, the explanation demonstrates that @=9 is a value that fits a valid solution derived from a plausible interpretation of the equations.