How can I solve 4 equations with 4 unknowns?

[malsch]

Homework Statement

First of all hi,

I was working some mathematics and ended up with 4 unknowns with 4 equations. I tried substituting equations but ended up with squared unknowns. The following is what i ended up with so far.

The Attempt at a Solution

Any help on how to solve this would be greatly appreciated.

Thank you.

 

[tiny-tim]

welcome to pf!

hi malsch! welcome to pf!

essentially, you have two equations of the form Px² + Qxy + Ry² = S

(i don't know whether this is the quickest way , but …)

i'd make it more symmetrical by getting rid of the xy terms, giving

x²/a² + y/b² = 1​

so x = acosθ, y = bsinθ,

and then use standard trigonometric identities to arrive at an equation in cos2θ and sin2θ

(it it's x²/a² minus y/b² = C², use cosh and sinh instead of cos and sin)

 

[malsch]

hi. thanks for your reply tiny-tim :)

i didn't understand how you can get rid of the xy terms and end up with x2/a2 + y/b2 = C2.

10x again

 

[malsch]

now that i come to it, i am guessing i can do the following:

Px2 + Qxy + Ry2 = S

=> (xA + yB)(xC + yD) = S

and find the values of x and y simultaneausly

 

[tiny-tim]

hi malsch!

(try using the X² icon just above the Reply box )

now that i come to it, i am guessing i can do the following:

Px2 + Qxy + Ry2 = S

=> (xA + yB)(xC + yD) = S

and find the values of x and y simultaneausly

i don't see how that helps

i didn't understand how you can get rid of the xy terms and end up with x2/a2 + y/b2 = C2.

eliminate the xy terms (in your case, the CG terms) …

what do you get? ​

 

[malsch]

yes you are right, that doesn't help :(.

You mean dividing through out by xy (CG in my case)?

 

[tiny-tim]

no, you have two equations with xy in …

combine them in such a way that there's no xy

(for example, put xy on the LHS in one, then substitute the RHS for xy in the other)

 

[malsch]

ok now i came up with:

4.5*10^-7 = -G² + 98,652.1C²

I was wondering how you can convert this to cos and sin.

And by doing so, you would end up by an unknown angle, right?

 

[tiny-tim]

ok now i came up with:

4.5*10^-7 = -G² + 98,652.1C²

I was wondering how you can convert this to cos and sin.

And by doing so, you would end up by an unknown angle, right?

that minus means you'll need to use cosh and sinh instead …

divide by a constant to get it into the form

C²/a² - G²/b² = 1​

and then tanh = b/a

(btw, i got the formula wrong in my first post , the RHS has to be 1, I've edited it now)

 

[malsch]

thats ok hehe. i found the angle using tanh = b/a.

now i am working with these formulas:

C = acoshθ, G = bsinhθ,

but how did you conclude to them? i know cosθ + sinθ = 1, and coshθ - sinhθ = 1

 

[Char. Limit]

thats ok hehe. i found the angle using tanh = b/a.

now i am working with these formulas:

C = acoshθ, G = bsinhθ,

but how did you conclude to them? i know cosθ + sinθ = 1, and coshθ - sinhθ = 1

Those last two formulas aren't true. You're missing a few things, namely ²'s.

[tex]cos^2(\theta) + sin^2(\theta)=1[/tex]

[tex]cosh^2(\theta) - sinh^2(\theta) = 1[/tex]

 

[malsch]

yes yes you are right. ok i think i know how to continue from here.

Thanks Char. Limit and Tiny Tim especially ;)