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What is the value of \(y\)?
(1) \(y^3 + 2y = y + 2y^2\)
(2) \(y^2 = y\)
(1) \(y^3 + 2y = y + 2y^2\)
(2) \(y^2 = y\)
16 Sep 2014, 01:10
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Official Solution:
What is the value of \(y\)?
(1) \(y^3+2y=y+2y^2\).
Rearrange the equation and factor out \(y\) to obtain \(y(y^2-2y+1)=0\), which is the same as \(y(y-1)^2=0\). Thus, \(y=0\) or \(y=1\). Not sufficient.
Note that we cannot divide \(y^3+2y=y+2y^2\) by \(y\) because \(y\) can be 0 and division by zero is not allowed. By dividing by \(y\), we would be incorrectly assuming that \(y\) does not equal zero, potentially excluding a valid solution (observe that \(y=0\) satisfies the equation). As a rule, never reduce an equation by a variable (or by an expression containing a variable) if you are not certain that the variable (or expression with the variable) does not equal zero. Remember, we cannot divide by zero.
(2) \(y^2=y\).
Rearrange the equation and factor out \(y\) to obtain \(y(y-1)=0\). We obtain the same two solutions: \(y=0\) or \(y=1\). Not sufficient.
(1)+(2) Combining the two statements does not provide any new information. Both statements result in the same possible solutions, \(y=0\) or \(y=1\).Not sufficient.
Answer: E
What is the value of \(y\)?
(1) \(y^3+2y=y+2y^2\).
Rearrange the equation and factor out \(y\) to obtain \(y(y^2-2y+1)=0\), which is the same as \(y(y-1)^2=0\). Thus, \(y=0\) or \(y=1\). Not sufficient.
Note that we cannot divide \(y^3+2y=y+2y^2\) by \(y\) because \(y\) can be 0 and division by zero is not allowed. By dividing by \(y\), we would be incorrectly assuming that \(y\) does not equal zero, potentially excluding a valid solution (observe that \(y=0\) satisfies the equation). As a rule, never reduce an equation by a variable (or by an expression containing a variable) if you are not certain that the variable (or expression with the variable) does not equal zero. Remember, we cannot divide by zero.
(2) \(y^2=y\).
Rearrange the equation and factor out \(y\) to obtain \(y(y-1)=0\). We obtain the same two solutions: \(y=0\) or \(y=1\). Not sufficient.
(1)+(2) Combining the two statements does not provide any new information. Both statements result in the same possible solutions, \(y=0\) or \(y=1\).Not sufficient.
Answer: E
