An algebra problem by Armain Labeeb

Algebra Level 2

Cecilia attempts to prove that \(0\,=\,1\). This was her proof:

\[\begin{align} -20 & \, \, =\, \, -20 \\ 16\, -\, 36 & \, \, =\, \, 25\, -\, 45 \\ 4^{ 2 }\, -\, 4\, \cdot \, 9 & \, \, =\, \, 5^{ 2 }\, -\, 5\, \cdot \, 9 \\ 4^{ 2 }\, -\, 4\, \cdot \, 9\, +\, \frac { 81 }{ 4 } & \, \, =\, \, 5^{ 2 }\, -\, 5\, \cdot \, 9\, +\, \frac { 81 }{ 4 } \\ 4^{ 2 }\, -\, 2\, \cdot \, 4\, \cdot \, \frac { 9 }{ 2 } \, +\, \left( \frac { 9 }{ 2 } \right) ^{ 2 }\, \, & \, \, =\, \, 5^{ 2 }\, -\, 5\, \cdot \, 9\, +\, \, \left( \frac { 9 }{ 2 } \right) ^{ 2 } \\ \left( 4\, -\, \frac { 9 }{ 2 } \right) ^{ 2 }\, \, & =\, \, \left( 5\, -\, \frac { 9 }{ 2 } \right) ^{ 2 } \\ \left( 4\, -\, \frac { 9 }{ 2 } \right) \, \, & =\, \, \left( 5\, -\, \frac { 9 }{ 2 } \right) \\ \left( 4\, -\, \frac { 9 }{ 2 } \right) \, +\, \frac { 1 }{ 2 } \, \, & =\, \, \left( 5\, -\, \frac { 9 }{ 2 } \right) \, +\, \frac { 1 }{ 2 } \\ 0 & \, \, =\, \, 1 \end{align} \]

Which is the first step where she made a mistake? Assume that the first line of the equation (\(-20=-20\)) is the first step.

Signs: \(a\,\cdot\,b\,\cdot\,c\,=\,(a\,\times\,b\,\times\,c)\).

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