2023 AMC 12A Problems/Problem 10
Contents
Problem
Positive real numbers and satisfy and . What is ?
Solution 1
Because , set , (). Put them in we get which implies . Solve the equation to get or . Since and are positive, and .
~plasta
Solution 2
Let's take the second equation and square root both sides. This will obtain . Solving the case where , we'd find that . This is known to be false because both and have to be positive, and implies that at least one of the variables is not positive. So we instead solve the case where . This means that . Inputting this value into the first equation, we find: This means that . Therefore,
~lprado
Solution 3: Quadratic formula
first expand
consider a=1 b=-2y c=-3y^2
or
and
and
we can see both x and y will be postive in
now do same as solution 2 This means that . Therefore,
Solution 4: Substitution
Since , we can rewrite the second equation as
Let . The second equation becomes
Substituting this into the first equation, we have
so .
Hence and
-Benedict T (countmath1)
Solution 5: Difference of Squares
We will use the difference of squares in the second equation.
Since x and y are positive, x+y is non-zero. Thus, .
Substituting into the first equation:
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See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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All AMC 12 Problems and Solutions |
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