If
\n@ + * = 16
\n+ # – & = 12
\n@ – & = 16
\nthen which one of the following is correct?<\/p>\n
[amp_mcq option1=”@ = 9″ option2=”* = 7″ option3=”# = 19″ option4=”& = 6″ correct=”option1″]<\/p>\n
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`.<\/p>\n
Let’s analyze the system assuming the second equation is `# – & = 12`:
\n1) @ + * = 16
\n2) # – & = 12
\n3) @ – & = 16<\/p>\n
From equation (3), we can express @ in terms of &: @ = 16 + &.
\nFrom equation (2), we can express # in terms of &: # = 12 + &.
\nSubstitute @ = 16 + & into equation (1):
\n(16 + &) + * = 16
\n16 + & + * = 16
\n& + * = 0
\nThis implies * = -&.<\/p>\n
The relationships between the variables are:
\n@ = 16 + &
\n* = -&
\n# = 12 + &<\/p>\n
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:<\/p>\n
A) @ = 9:
\nIf @ = 9, then from @ = 16 + &, we get 9 = 16 + & => & = 9 – 16 = -7.
\nNow find * and # using & = -7:
\n* = -& = -(-7) = 7.
\n# = 12 + & = 12 + (-7) = 5.
\nLet’s check if the values (@=9, *=7, #=5, &=-7) satisfy the original equations:
\n1) @ + * = 9 + 7 = 16 (Correct)
\n2) # – & = 5 – (-7) = 5 + 7 = 12 (Correct, assuming this form of eq 2)
\n3) @ – & = 9 – (-7) = 9 + 7 = 16 (Correct)
\nSo, the solution (@=9, *=7, #=5, &=-7) is valid if the second equation is `# – & = 12`. In this solution, the statement @=9 is true.<\/p>\n
Let’s check other options against the relationships @ = 16 + &, * = -&, # = 12 + &:
\nB) * = 7:
\nIf * = 7, then from * = -&, we get 7 = -& => & = -7.
\nThis gives the same value for & as option A, leading to the same solution (@=9, *=7, #=5, &=-7). In this solution, * = 7 is also true.<\/p>\n
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.<\/p>\n
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`).<\/p>\n
Let’s also consider the alternative interpretation where the second equation is `* + # – & = 12`:
\n1) @ + * = 16
\n2) * + # – & = 12
\n3) @ – & = 16
\nFrom (1) and (3), subtracting (3) from (1) gives: (@ + *) – (@ – &) = 16 – 16 => * + & = 0 => * = -&.
\nSubstitute * = -& into (1): @ + (-&) = 16 => @ – & = 16, which is (3).
\nSubstitute * = -& into (2): (-&) + # – & = 12 => # – 2& = 12 => # = 12 + 2&.
\nRelationships: @ = 16 + &, * = -&, # = 12 + 2&.
\nA) @ = 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.
\nB) * = 7: 7 = -& => & = -7. (@=9, *=7, #=-2, &=-7). * = 7 is also true in this solution.<\/p>\n