Only one?

Algebra Level 2

Real numbers x x , y y , and z z are such that x y z = 1 xyz = 1 . Find the value of the expression below.

1 x y + x + 1 + y y z + y + 1 + 1 x y z + y z + y \frac{1}{xy+x+1} + \dfrac{y}{yz+y+1} + \dfrac{1}{xyz+yz+y}


The answer is 1.

Tin Le by Tin Le , 15, Vietnam

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2 solutions

Chew-Seong Cheong
Apr 14, 2019

A = 1 x y + x + 1 + y y z + y + 1 + 1 x y z + y z + y Given that x y z = 1 = 1 x y + x + 1 + x y x y z + x y + x + x x ⋅ 1 + x y z + x y Multiply the last two fractions by x x = 1 x y + x + 1 + x y 1 + x y + x + x x + 1 + x y = x y + x + 1 x y + x + 1 = 1 \begin{aligned} A & = \frac 1{xy+x+1} + \frac y{yz+y+1} + \frac 1{{\color{#3D99F6}xyz}+yz+y} & \small \color{#3D99F6} \text{Given that }xyz = 1 \\ & = \frac 1{xy+x+1} + \frac {{\color{#D61F06}x}y}{{\color{#D61F06}x}yz+{\color{#D61F06}x}y+{\color{#D61F06}x}} + \frac {\color{#D61F06}x}{{\color{#D61F06}x}\cdot{\color{#3D99F6}1}+{\color{#D61F06}x}yz+{\color{#D61F06}x}y} & \small \color{#D61F06} \text{Multiply the last two fractions by }\frac xx \\ & = \frac 1{xy+x+1} + \frac {xy}{1+xy+x} + \frac x{x+1+xy} \\ & = \frac {xy+x+1}{xy+x+1} \\ & = \boxed 1 \end{aligned}

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@Tin Le, sorry but it is not natural to start an English sentence with small (lower case) letter "x", therefore, I have changed it to start with "Real numbers..." It is necessary that all mathematical quantities such as variables x x , y y , and z z and equation x y z = 1 xyz=1 be in LaTex. The separator for variables in English is comma (,) and not semi-colon (;). Thanks,

Chew-Seong Cheong - 2 years, 1 month ago

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Thank you, I appreciate your note.

Tin Le - 2 years, 1 month ago

Names may begin sentences. Names should be in their native case, e. g., e. e. cummings . In the case of multi-lists in English, both semicolon and commas are used as separators.

A Former Brilliant Member - 2 years, 1 month ago

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Thanks for your comments. I edited the problem and was explaining to @Tin Le , why I amended it. Of course, nouns can start a sentence. "Real numbers" above is a noun. It was started with a lowercase x x which can be avoided by rephrasing. Of course semicolons are used in separating multiple lists with commas but only one list appears here.

Chew-Seong Cheong - 2 years, 1 month ago

x x , y y and z z must be all non-zero to allow x y z = 1 x\ y\ z=1 to be true. Subsituting for x x the expression 1 y z \frac{1}{y z} and simplifying the resultant expression gives 1 1 .

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