Four Non-Linear Simultaneous Equations

In summary, Sudharaka was able to solve all 4 equations by hand. Maxima was only able to approximate the first equation.
  • #1
Wilmer
307
0
4 equations, 4 unknowns:

\[\frac{u(r^2 - u^2)}{r^2 + u^2}=156~~~~~~~~~~(1)\]
\[\frac{v(r^2 - v^2)}{r^2 + v^2} = 96~~~~~~~~~(2)\]
\[\frac{w(r^2 - w^2)}{r^2 + w^2} = 63~~~~~~~~~~(3)\]
\[\frac{315uvw + 24336vw + 9216uw + 3969uv}{2r} = 943488~~~~~~~~~(4)\]

Who can solve that mess?
 
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  • #2
Wilmer said:
4 equations, 4 unknowns:

u(r^2 - u^2) / (r^2 + u^2) = 156 [1]
v(r^2 - v^2) / (r^2 + v^2) = 96 [2]
w(r^2 - w^2) / (r^2 + w^2) = 63 [3]
(315uvw + 24336vw + 9216uw + 3969uv) / (2r) = 943488 [4]

Who can solve that mess?

Hi Wilmer, :)

If you are only concerned about real roots these are the solutions Maxima gives. Note that the first one is only an approximate.\[u=-166.2623906705539,v=-110.8238636363636,r=-29.66972878390201,w=-81.98732171156894\]

\[u=260,v=104,r=520,w=65\]

Kind Regards,
Sudharaka.
 
  • #3
Sudharaka said:
\[u=260,v=104,r=520,w=65\]
Hi Suds!
Yes, that's correct: I had this one solved,
but by brute force.

All I'm trying to do is find a way to solve
this "by hand".
 
  • #4
Wilmer said:
Hi Suds!
Yes, that's correct: I had this one solved,
but by brute force.

All I'm trying to do is find a way to solve
this "by hand".
I think that I may have been part way towards finding Sudharaka's solution "by hand". I was looking at the function $f(x) = \dfrac{x(1-x^2)}{1+x^2}.$ For positive values of $x$, this has a maximum value of 3/10, which occurs when $x=1/2.$

Your equation (1) says that $f(u/r) = 156/r.$ This tells you that $156/r\leqslant 3/10$, or $r\geqslant 520.$ Also, the value $r=520$ can only occur if $u=520/2=260$. I was going to explore this further, to see if there were values of $v$ and $w$ compatible with those values of $r$ and $u$, but Sudharaka got there first.
 
  • #5
Background info, in case useful:
Code:
              C
 
 
        D                     E
              U          V                    
                   M
                   
                   W
                   
B                  F                        A
Acute triangle ABC: M is circumcenter.
U, V and W are the incenters of triangles BCM, ACM and ABM respectively:
and DM, EM and FM are the perpendicular heights.

NOT GIVENS: a = BC = 624, b = AC = 960, c = AB = 1008, r = 520 = AM=BM=CM.
NOT GIVENS: u = UM = 260, v = VM = 104, w = WM = 65.
GIVENS: d = DU = 156, e = EV = 96, f = FW = 63.

Work to set up the 4 equations:
a = 2dr / u , b = 2er / b , c = 2fr / w

from triangleBCM: u(r^2 - u^2) / (r^2 + u^2) = d [1]
from triangleACM: v(r^2 - v^2) / (r^2 + v^2) = e [2]
from triangleABM: w(r^2 - w^2) / (r^2 + w^2) = f [3]

area(BCM + ACM + ABM) = areaABC; leads to :
[uvw(d + e + f) + vwd^2 + uwe^2 + uvf^2] / (2r) = def [4]

Inserting the givens gives us:
u(r^2 - u^2) / (r^2 + u^2) = 156 [1]
v(r^2 - v^2) / (r^2 + v^2) = 96 [2]
w(r^2 - w^2) / (r^2 + w^2) = 63 [3]
(315uvw + 24336vw + 9216uw + 3969uv) / (2r) = 943488 [4]

I'm simply curious as to the possibility of solving these 4 simultaneous equations.
 

FAQ: Four Non-Linear Simultaneous Equations

What are four non-linear simultaneous equations?

Four non-linear simultaneous equations are a set of four equations that involve variables raised to a power or multiplied together. These equations cannot be solved using traditional algebraic methods and often require the use of graphical or numerical methods.

How do you solve four non-linear simultaneous equations?

There are several methods for solving four non-linear simultaneous equations. One method is to graph the equations and find the points of intersection, which will be the solutions. Another method is to use substitution, where you solve one equation for one variable and substitute it into the other equations. You can also use elimination, where you multiply one or more equations by a constant to eliminate a variable. Lastly, you can use numerical methods, such as the Newton-Raphson method, to approximate the solutions.

Why are four non-linear simultaneous equations difficult to solve?

Four non-linear simultaneous equations are difficult to solve because they involve variables raised to a power or multiplied together, making them non-linear. This means that the traditional algebraic methods used to solve linear equations cannot be applied. Additionally, these equations often have multiple solutions, making it challenging to find the correct solution.

What are some real-life applications of four non-linear simultaneous equations?

Four non-linear simultaneous equations are commonly used in engineering and physics to model complex systems and phenomena. They can also be used in economics to analyze supply and demand, in chemistry to study chemical reactions, and in biology to model population growth and interactions.

What are some strategies for checking the accuracy of solutions to four non-linear simultaneous equations?

One strategy is to plug the solutions into each of the original equations and see if they satisfy all of them. Another strategy is to use a graphing calculator or software to graph the equations and see if the solutions correspond to the points of intersection. Additionally, you can use numerical methods to approximate the solutions and compare them to the solutions obtained through other methods.

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