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0
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I need help with this system:

$$\begin{eqnarray} x(y+z)&=&39\\ y(x+z)&=&60\\ z(x+y)&=&63 \\ x^2+y^2+z^2&=& \ ? \end{eqnarray}$$

What is x^2 + y^2 + z^2?

Jun 30, 2020

#1
+25555
+2

I need help with this system:
$$\begin{eqnarray} x(y+z)&=&39\\ y(x+z)&=&60\\ z(x+y)&=&63 \\ x^2+y^2+z^2&=& \ ? \end{eqnarray}$$

$$\begin{array}{|lrcll|} \hline & x(y+z)&=&39 \\ & xy+xz &=& 39 \\ & \mathbf{xz} &=& \mathbf{39-xy} \qquad (1) \\\\ & y(x+z)&=&60 \\ & yx+yz &=&60 \\ & \mathbf{yz} &=& \mathbf{60-xy} \qquad (2) \\ \hline (1)+(2): & xz+yz &=& 39-xy+60-xy \\ & z(x+y) &=& 99-2xy \quad | \quad \mathbf{z(x+y)=63} \\ & 63 &=& 99-2xy \\ & 2xy &=& 99-63 \\ & 2xy &=& 36 \\ & \mathbf{xy} &=& \mathbf{18} \quad \text{or} \quad \mathbf{y=\dfrac{18}{x}} \qquad (3) \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline x(y+z)&=& 39 \quad | \quad \mathbf{y=\dfrac{18}{x}} \\\\ x\left(\dfrac{18}{x}+z\right) &=& 39 \\\\ 18 +xz &=& 39 \\\\ xz &=& 39-18 \\\\ \mathbf{xz} &=& \mathbf{21} \quad \text{or} \quad \mathbf{x=\dfrac{21}{z}} \qquad (4) \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline z(x+y)&=& 63 \quad | \quad \mathbf{x=\dfrac{21}{z}} \\\\ z\left(\dfrac{21}{z}+y \right)&=& 63 \\\\ 21+zy &=& 63 \\\\ zy &=& 63-21 \\\\ \mathbf{zy} &=& \mathbf{42} \qquad (5) \\ \hline \end{array}$$

$$\begin{array}{|lrcll|} \hline \dfrac{(3)}{(5)}: & \dfrac{xy}{zy} &=& \dfrac{18}{42} \\\\ & \dfrac{x}{z} &=& \dfrac{3}{7} \quad | \quad \mathbf{x=\dfrac{21}{z}}\quad \text{or} \quad \mathbf{z=\dfrac{21}{x}} \\\\ & \dfrac{x}{\dfrac{21}{x}} &=& \dfrac{3}{7} \\\\ & \dfrac{x^2}{21} &=& \dfrac{3}{7} \\\\ & x^2 &=& \dfrac{3*21}{7} \\\\ & x^2 &=& 3*3 \\\\ & \mathbf{x} &=& \pm \mathbf{3} \\ \hline & \mathbf{y} &=& \mathbf{\dfrac{18}{x}} \\\\ & y &=& \dfrac{18}{\pm 3} \\\\ & \mathbf{y} &=& \mathbf{\pm 6 } \\ \hline & \mathbf{z} &=& \mathbf{\dfrac{21}{x} } \\\\ & z &=& \dfrac{21}{\pm 3} \\\\ & \mathbf{z} &=& \mathbf{\pm 7} \\ \hline & x^2+y^2+z^2 &=& (\pm 3)^2+(\pm 6)^2+(\pm 7)^2 \\\\ & &=& 9+36+49 \\\\ &\mathbf{ x^2+y^2+z^2} &=& \mathbf{94} \\ \hline \end{array}$$

Jul 1, 2020