y-2-1-x-2-4-z-2-9-10-x-y-z-

Question Number 142138 by mathdanisur last updated on 26/May/21

\sqrt{y^{2} + 1} + \sqrt{x^{2} + 4} + \sqrt{z^{2} + 9} = 10

x + y + z = ?

Commented by mathdanisur last updated on 27/May/21

c o o l s i r t h a n k y o u

Commented by mr W last updated on 27/May/21

\sqrt{y^{2} + 1^{2}} + \sqrt{x^{2} + 2^{2}} + \sqrt{z^{2} + 3^{2}} ⩾ \sqrt{{(x + y + z)}^{2} + {(1 + 2 + 3)}^{2}}

\Rightarrow {(x + y + z)}^{2} ⩽ 10^{2} - 6^{2} = 64 = 8^{2}

\Rightarrow - 8 ⩽ x + y + z ⩽ 8

Commented by MJS_new last updated on 26/May/21

no unique value

Commented by MJS_new last updated on 26/May/21

\sqrt{x^{2} + 4} = a \Leftrightarrow x = \pm \sqrt{a^{2} - 4}; 2 ⩽ a ⩽ 10

\sqrt{y^{2} + 1} = b \Leftrightarrow y = \pm \sqrt{b^{2} - 1}; 1 ⩽ b ⩽ 10

\sqrt{z^{2} + 9} = c \Leftrightarrow z = \pm \sqrt{c^{2} - 9}; 3 ⩽ c ⩽ 10

a + b + c = 10

infinite values for

x + y + z = \pm \sqrt{a^{2} - 4} \pm \sqrt{b^{2} - 1} \pm \sqrt{c^{2} - 9}

Commented by MJS_new last updated on 26/May/21

- 8 ⩽ x + y + z ⩽ 8

Commented by mathdanisur last updated on 29/May/21

c o o l S i r t h a n k s