Respuesta :
Answer:
5
Step-by-step explanation:
We would like to find out the value of [tex] x + y + z [/tex] from the given equations , which are ;
[tex]\longrightarrow \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0[/tex]
[tex]\longrightarrow \\ x^2+y^2+z^2=25[/tex]
Now consider ,
[tex]\longrightarrow \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\\[/tex]
[tex]\longrightarrow \dfrac{xy + yz + zx }{xyz }=0\\[/tex]
[tex]\longrightarrow xy + yz + zx = 0(xyz)\\ [/tex]
[tex]\longrightarrow xy + yz + zx = 0\\ [/tex]
[tex]\longrightarrow 2(xy + yz + zx)=0(2)\\[/tex]
[tex]\longrightarrow 2(xy + yz + zx)=0[/tex]
Now recall the identity ,
[tex]\longrightarrow (a + b + c)^2=a^2+b^2+c^2+2(ab + bc + ca)[/tex]
Plug in the values ,
[tex]\longrightarrow (x+y+z)^2= 25 + 0\\ [/tex]
[tex]\longrightarrow (x + y + z )^2=25\\[/tex]
[tex]\longrightarrow (x + y + z)=\sqrt{25}\\[/tex]
[tex]\longrightarrow \underline{\underline{\boldsymbol{ x + y + z = 5}}}{} [/tex]
And we are done !
Answer :
- 5
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Step-by-step explanation:
Given,
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[tex]{ \longrightarrow \qquad{ { \sf{ \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} = 0}}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \sf{ {x}^{2} + {y}^{2} + {z}^{2} = 25 }}}}[/tex]
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To Find :
[tex]{ \longrightarrow \qquad{ { \sf{ x + y + z }}}}[/tex]
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Solution :
[tex]{ \longrightarrow \qquad{ { \sf{ \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} = 0}}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \sf{ \dfrac{yz + zx + xy}{xyz} = 0}}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \sf{ {yz + zx + xy} = 0}}}}[/tex]
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Multiplying both sides by 2 :
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[tex]{ \longrightarrow \qquad{ { \sf{ 2( {yz + zx + xy} )= 0.2}}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \sf{ 2( {yz + zx + xy} )= 0}}}}[/tex]
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We know,
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[tex]{ \longrightarrow \qquad{ { \sf{ {(x + y + z)}^{2} = {x}^{2} + {y}^{2} + {z}^{2} + 2(xy + yz + zx) }}}}[/tex]
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As, It is given that,
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[tex]{ \longrightarrow \qquad{ { \sf{ {x}^{2} + {y}^{2} + {z}^{2} = 25 }}}}[/tex]
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and as we get,
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[tex]{ \longrightarrow \qquad{ { \sf{ 2( {yz + zx + xy} )= 0}}}} \\ \\{ \longrightarrow \qquad{ { \sf{ 2( {xy + yz + zx} )= 0}}}}[/tex]
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Now, We'll substitute it in the formula :
[tex]{ \longrightarrow \qquad{ { \sf{ {(x + y + z)}^{2} = 25 + 0 }}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \sf{ {(x + y + z)}^{2} = {25} }}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \sf{ {x + y + z}^{} = \sqrt{25} }}}}[/tex]
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[tex]{ \longrightarrow \qquad{ { \bf{ \pmb {x + y + z}^{} = {5} }}}}[/tex]
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Therefore,
- The value of x + y + z is 5 .
