x^2+y^2=z^2

y^2*(y+z)^2+x^2*y^2=128*(y+z)^2

x^2*(x+z)^2+x^2*y^2=81*(x+z)^2

asked by guest
on Sep 21, 2024 at 10:44 am



You asked:

Solve the system of equations:\[\begin{aligned}{x}^{2} + {y}^{2} &= {z}^{2}\\{y}^{2} \cdot {\left( y + z \right)}^{2} + {x}^{2} \cdot {y}^{2} &= 128 \cdot {\left( y + z \right)}^{2}\\{x}^{2} \cdot {\left( x + z \right)}^{2} + {x}^{2} \cdot {y}^{2} &= 81 \cdot {\left( x + z \right)}^{2}\end{aligned}\]

MathBot Answer:

The system of equations has \(7\) solutions.

\[x = 0, y = 0, z = 0\]\[x = - \frac{18 \sqrt{5}}{5}, y = - \frac{24 \sqrt{5}}{5}, z = - 6 \sqrt{5}\]\[x = \frac{18 \sqrt{5}}{5}, y = \frac{24 \sqrt{5}}{5}, z = 6 \sqrt{5}\]\[x = \frac{9 \left(- 19 \sqrt{46} + 119 \sqrt{2} i\right)}{\sqrt{-43 + \sqrt{23} i} \left(\sqrt{23} + 43 i\right)}, y = \frac{4 \sqrt{-43 + \sqrt{23} i} \left(- \sqrt{46} - 7 \sqrt{2} i\right)}{\sqrt{23} + 43 i}, z = \sqrt{- \frac{43}{2} + \frac{\sqrt{23} i}{2}}\]\[x = \frac{9 \cdot \left(19 \sqrt{46} - 119 \sqrt{2} i\right)}{\sqrt{-43 + \sqrt{23} i} \left(\sqrt{23} + 43 i\right)}, y = \frac{4 \sqrt{-43 + \sqrt{23} i} \left(\sqrt{46} + 7 \sqrt{2} i\right)}{\sqrt{23} + 43 i}, z = - \sqrt{- \frac{43}{2} + \frac{\sqrt{23} i}{2}}\]\[x = \frac{9 \cdot \left(19 \sqrt{46} + 119 \sqrt{2} i\right)}{\sqrt{-43 - \sqrt{23} i} \left(\sqrt{23} - 43 i\right)}, y = \frac{4 \sqrt{-43 - \sqrt{23} i} \left(\sqrt{46} - 7 \sqrt{2} i\right)}{\sqrt{23} - 43 i}, z = - \sqrt{- \frac{43}{2} - \frac{\sqrt{23} i}{2}}\]\[x = \frac{9 \cdot \left(19 \sqrt{46} + 119 \sqrt{2} i\right)}{\sqrt{-43 - \sqrt{23} i} \left(- \sqrt{23} + 43 i\right)}, y = \frac{4 \sqrt{-43 - \sqrt{23} i} \left(- \sqrt{46} + 7 \sqrt{2} i\right)}{\sqrt{23} - 43 i}, z = \sqrt{- \frac{43}{2} - \frac{\sqrt{23} i}{2}}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

+ Submit Question