+0  
 
0
79
1
avatar+542 

 

Find an ordered triple (x,y,z) of real numbers satisfying x<= y<= z and the system of equations
sqrtx + sqrty +sqrtz = 10

x+y+z=38

(sqrtxyz) = 30

waffles  Jan 25, 2018
Sort: 

1+0 Answers

 #1
avatar+18956 
+2

Find an ordered triple (x,y,z) of real numbers satisfying x<= y<= z and the system of equations
sqrtx + sqrty +sqrtz = 10
x+y+z=38
(sqrtxyz) = 30

 

\(\begin{array}{|lrcll|} \hline (1) & \sqrt{x} + \sqrt{y} +\sqrt{z} &=& 10 \\ (2) & x+y+z &=& 38 \\ (3) & \sqrt{xyz} &=& 30 \\ \hline \end{array}\)

 

1.

\(\begin{array}{|lrcll|} \hline & (\sqrt{x} + \sqrt{y} +\sqrt{z})^2 &=& 10^2 \\ & \underbrace{x+y+z}_{=38} +2( \sqrt{xy}+\sqrt{xz}+\sqrt{yz} ) &=& 100 \\ & 38 +2( \sqrt{xy}+\sqrt{xz}+\sqrt{yz} ) &=& 100 \quad & | \quad -38 \\ & 2( \sqrt{xy}+\sqrt{xz}+\sqrt{yz} ) &=& 62 \quad & | \quad :2 \\ \mathbf{(4)}& \mathbf{\sqrt{xy}+\sqrt{xz}+\sqrt{yz}} &\mathbf{=}& \mathbf{31} \\ \hline \end{array}\)

 

\(\begin{array}{lcll} \text{We substitute:} \\ \quad x_1 = \sqrt{x} & \text{ or }& x = x_1^2 \\ \quad x_2 = \sqrt{y} & \text{ or }& y = x_2^2 \\ \quad x_3 = \sqrt{z} & \text{ or }& z = x_3^2 \\ \end{array}\)

 

\(\begin{array}{|rcll|} \hline x_1+x_2+x_3 &=& \sqrt{x} + \sqrt{y} +\sqrt{z} \\ &=& 10 \\ x_1\cdot x_2 \cdot x_3 &=& \sqrt{xyz} \\ &=& 30 \\ x_1\cdot x_2 + x_1 \cdot x_3 + x_2 \cdot x_3 &=& \sqrt{xy}+\sqrt{xz}+\sqrt{yz} \\ &=& 31 \\ \hline \end{array}\)

 

 

\(\begin{array}{lcll} \text{We set:} \\ \quad p &=& -(x_1+x_2+x_3) \\ &=& -10 \\ \quad q &=& x_1\cdot x_2 + x_1 \cdot x_3 + x_2 \cdot x_3 \\ &=& 31 \\ \quad r &=& -(x_1\cdot x_2 \cdot x_3) \\ &=& -30 \\ \end{array}\)

 

\(\begin{array}{lrcll} \text{The cubic equation:} \\ \boxed{ x^3 - 10x^2+31x-30 = 0 } \\ \end{array}\)

 

The first solution is \(x_1 = 2\)

 

Long division:

\(\begin{array}{rcll} x^2-8x+15 = (x-3)(x-5) \end{array}\)

so \(x_2 = 3\) and \(x_3 = 5\)

 

\(\begin{array}{|rcll|} \hline x_1 = 2 & x &=& x_1^2 \\ & \mathbf{x} &\mathbf{=}& \mathbf{4} \\\\ x_2 = 3 & y &=& x_2^2 \\ & \mathbf{y} &\mathbf{=}& \mathbf{9} \\\\ x_3 = 5 & z &=& x_3^2 \\ & \mathbf{z} &\mathbf{=}& \mathbf{25} \\ \hline \end{array}\)

 

The ordered triple (4, 9, 25) of real numbers satisfying \(4\le 9\le 25\) and the system of equations.

 

laugh

heureka  Jan 25, 2018

12 Online Users

avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details