System of Equations | Mathematics | Page 1

gray windBOT Dec 22, 2023, 1:37 PM

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unreal swallow Dec 22, 2023, 2:56 PM

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partially differentiate both the equations w.r.t x and y

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you will get 2 linear equations each

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solve the linear equations

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you will get value of x and y for both the equations

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or use quadratic formula

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for x² + (3y)x + (2y²-6) =0

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you will get two linear equation

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solve them, you will get the answer

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yes

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then also do partial derivation w.r.t y

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you will get 2 eqn

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this method works for general equation of second degree

unreal swallow Dec 22, 2023, 3:05 PM

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unreal swallow for x² + (3y)x + (2y²-6) =0

or use quadratic formula

frosty furnaceBOT Dec 22, 2023, 3:05 PM

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@untold bronze has given 1 rep to @unreal swallow

unreal swallow Dec 22, 2023, 3:28 PM

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@raw shoal sorry for wasting your time

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but my method wont work here

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partial differentiation methods works for pair of straight lines

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and i guess

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this has no solution

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the first eqn

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the lines will never intersect though

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ahh, so the second eqn will intersect the first

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indeed, this is new for me.

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sorry

unreal swallow Dec 22, 2023, 3:48 PM

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yes, till now i have read bout pair of straight lines, i will get to study this when i reach 12th and study more about conics

remote lynx Dec 22, 2023, 4:02 PM

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unreal swallow yes, till now i have read bout pair of straight lines, i will get to study this ...

but if we solve this from a purely algebraic perspective...

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i wonder how long is long

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cause i was able to solve it in one page... is that an improvement?

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$x^2 + 3xy + 2y^2 = 6 \ 2x^2 + xy + 5y^2 = 8$

finite rockBOT Dec 22, 2023, 4:04 PM

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Hariharan

remote lynx Dec 22, 2023, 4:04 PM

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double the first equation and subtract it from the second one... youll get 👇

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$y^2 - 5xy + 4 = 0 \ \Rightarrow x = \frac{y^2 + 4}{5y}$

finite rockBOT Dec 22, 2023, 4:05 PM

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Hariharan

remote lynx Dec 22, 2023, 4:05 PM

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and we substitute this back in the first equation (given) 👇

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$\left(\frac{y^2 + 4}{5y}\right)^2 + 3y\left(\frac{y^2 + 4}{5y}\right) + 2y^2 = 6$

finite rockBOT Dec 22, 2023, 4:07 PM

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Hariharan

remote lynx Dec 22, 2023, 4:08 PM

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this looks scary at first but multiplying throughout with $25y^2$ will make this an equation of degree 4... but it has only even powers. so we can consider it as a quadratic equation in $y^2$ and apply the quadratic formula and you should be fine

finite rockBOT Dec 22, 2023, 4:08 PM

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Hariharan

remote lynx Dec 22, 2023, 4:09 PM

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it wil give $33y^4 - 71y^2 + 18 = 0$

finite rockBOT Dec 22, 2023, 4:09 PM

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Hariharan

frosty furnaceBOT Dec 22, 2023, 4:12 PM

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@untold bronze has given 1 rep to @remote lynx

remote lynx Dec 22, 2023, 4:12 PM

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yep

remote lynx Dec 22, 2023, 4:13 PM

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remote lynx this looks scary at first but multiplying throughout with $25y^2$ will make this...

provided... both the solutions for y^2 from the quadratic equation here are positive

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there probably is, but this is the only one i could think of

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probably not the best, but it gets the job done (hopefully)

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do match these answers with the ones from wolfram... cause maybe i made a mistake

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you can close the thread if youre done, or you can wait :)

#System of Equations