25 + x^2 - 10*x*y = 16, 64 + x^2 -16*x*y = (32/5)^2

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on Jan 31, 2025 at 12:28 am



You asked:

Solve the system of equations:25+x210xy=1664+x216xy=(325)2\begin{aligned}25 + {x}^{2} - 10 x y &= 16\\64 + {x}^{2} - 16 x y &= {\left( \frac{32}{5} \right)}^{2}\end{aligned}

MathBot Answer:

The system of equations has 22 solutions.

x=6105,y=3910200x = - \frac{6 \sqrt{10}}{5}, y = - \frac{39 \sqrt{10}}{200}x=6105,y=3910200x = \frac{6 \sqrt{10}}{5}, y = \frac{39 \sqrt{10}}{200}


Solve x210xy+25=16x^{2} - 10 x y + 25 = 16 for xx. x=5y25y29,x=5y+25y29x = 5 y - \sqrt{25 y^{2} - 9}, x = 5 y + \sqrt{25 y^{2} - 9}Substitute 5y25y295 y - \sqrt{25 y^{2} - 9} for xx in x216xy+64=102425x^{2} - 16 x y + 64 = \frac{1024}{25} and simplify. x216xy+64=102425(5y25y29)216(5y25y29)y+64=10242530y2+6y25y29+55=102425\begin{aligned}x^{2} - 16 x y + 64 &= \frac{1024}{25} \\ \left(5 y - \sqrt{25 y^{2} - 9}\right)^{2} - 16 \left(5 y - \sqrt{25 y^{2} - 9}\right) y + 64 &= \frac{1024}{25} \\ - 30 y^{2} + 6 y \sqrt{25 y^{2} - 9} + 55 &= \frac{1024}{25} \end{aligned}Substitute 3910200- \frac{39 \sqrt{10}}{200} into x210xy+25=16x^{2} - 10 x y + 25 = 16 to solve for xx. x210(3910200)x+25=16x2+3910x20+25=16x2+3910x20+9=020x2+3910x+18020=0x=6105,x=3104\begin{aligned}x^{2} - 10 \left(- \frac{39 \sqrt{10}}{200}\right) x + 25 &= 16 \\ x^{2} + \frac{39 \sqrt{10} x}{20} + 25 &= 16 \\x^{2} + \frac{39 \sqrt{10} x}{20} + 9 &= 0 \\ \frac{20 x^{2} + 39 \sqrt{10} x + 180}{20} &= 0 \\ x = - \frac{6 \sqrt{10}}{5}&, x = - \frac{3 \sqrt{10}}{4}\end{aligned}This yields the following solution. x=6105,y=3910200\begin{aligned}x = - \frac{6 \sqrt{10}}{5},\,y = - \frac{39 \sqrt{10}}{200}\end{aligned}Substitute 5y25y295 y - \sqrt{25 y^{2} - 9} for xx in x216xy+64=102425x^{2} - 16 x y + 64 = \frac{1024}{25} and simplify. x216xy+64=102425(5y25y29)216(5y25y29)y+64=10242530y2+6y25y29+55=102425\begin{aligned}x^{2} - 16 x y + 64 &= \frac{1024}{25} \\ \left(5 y - \sqrt{25 y^{2} - 9}\right)^{2} - 16 \left(5 y - \sqrt{25 y^{2} - 9}\right) y + 64 &= \frac{1024}{25} \\ - 30 y^{2} + 6 y \sqrt{25 y^{2} - 9} + 55 &= \frac{1024}{25} \end{aligned}Substitute 5y+25y295 y + \sqrt{25 y^{2} - 9} for xx in x216xy+64=102425x^{2} - 16 x y + 64 = \frac{1024}{25} and simplify. x216xy+64=102425(5y+25y29)216(5y+25y29)y+64=10242530y2+6y25y2955=102425\begin{aligned}x^{2} - 16 x y + 64 &= \frac{1024}{25} \\ \left(5 y + \sqrt{25 y^{2} - 9}\right)^{2} - 16 \left(5 y + \sqrt{25 y^{2} - 9}\right) y + 64 &= \frac{1024}{25} \\ 30 y^{2} + 6 y \sqrt{25 y^{2} - 9} - 55 &= - \frac{1024}{25} \end{aligned}Substitute 3910200\frac{39 \sqrt{10}}{200} into x210xy+25=16x^{2} - 10 x y + 25 = 16 to solve for xx. x2103910200x+25=16x23910x20+25=16x23910x20+9=020x23910x+18020=0x=3104,x=6105\begin{aligned}x^{2} - 10 \cdot \frac{39 \sqrt{10}}{200} x + 25 &= 16 \\ x^{2} - \frac{39 \sqrt{10} x}{20} + 25 &= 16 \\x^{2} - \frac{39 \sqrt{10} x}{20} + 9 &= 0 \\ \frac{20 x^{2} - 39 \sqrt{10} x + 180}{20} &= 0 \\ x = \frac{3 \sqrt{10}}{4}&, x = \frac{6 \sqrt{10}}{5}\end{aligned}This yields the following solution. x=6105,y=3910200\begin{aligned}x = \frac{6 \sqrt{10}}{5},\,y = \frac{39 \sqrt{10}}{200}\end{aligned}

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