Knowing your x , y , z x, y, z

Algebra Level 3

Given:

{ x + 1 y = 5 y + 1 z = 12 z + 1 x = 13 \begin{cases} x+\dfrac{1}{y}=5 \\ y+\dfrac{1}{z}=12 \\ z+\dfrac{1}{x}=13 \end{cases}

then what is the value of x y z + 1 x y z ? \large xyz+\frac{1}{xyz}?


The answer is 750.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

David Vreken
Nov 10, 2018

5 ⋅ 12 ⋅ 13 = ( x + 1 y ) ( y + 1 z ) ( z + 1 x ) 5 \cdot 12 \cdot 13 = (x + \frac{1}{y})(y + \frac{1}{z})(z + \frac{1}{x})

5 ⋅ 12 ⋅ 13 = x y z + x + y + z + 1 x + 1 y + 1 z + 1 x y z 5 \cdot 12 \cdot 13 = xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz}

5 ⋅ 12 ⋅ 13 = x y z + 1 x y z + ( x + 1 y ) + ( y + 1 z ) + ( z + 1 x ) 5 \cdot 12 \cdot 13 = xyz + \frac{1}{xyz} + (x + \frac{1}{y}) + (y + \frac{1}{z}) + (z + \frac{1}{x})

5 ⋅ 12 ⋅ 13 = x y z + 1 x y z + 5 + 12 + 13 5 \cdot 12 \cdot 13 = xyz + \frac{1}{xyz} + 5 + 12 + 13

x y z + 1 x y z = 5 ⋅ 12 ⋅ 13 − ( 5 + 12 + 13 ) xyz + \frac{1}{xyz} = 5 \cdot 12 \cdot 13 - (5 + 12 + 13)

x y z + 1 x y z = 750 xyz + \frac{1}{xyz} = \boxed{750}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you, nice solution.

Hana Wehbi - 2 years, 7 months ago

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