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# needs to solve system of 4 eqns, 4 unknowns

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## needs to solve system of 4 eqns, 4 unknowns

(OP)
Hello All,

I what what I thought (once again!) was a fairly straight-forward problem to solve, but I can't quite seem to get there. I have 4 eqns, 4 unknowns. Equations are along the lines of:

sqrt(x^2+y^2) + z + sqrt((val1-x)^2 + (y-val2)) + Const1 = 0

sqrt(x^2+y^2) + z + Const2 = 0

sqrt(x^2+y^2) + z + 3*sqrt((x-val3)^2 + (y-val3)) + Const3 = 0

sqrt(x^2+y^2) + z + 2*sqrt((val4-x)^2 + (y-val2)) + Const4 = 0

In each case, "valn" is known, as is "Constn." "Constn" may consist of 1 or more values, here combined for simplicity.

I've tried a number of methods...from transforming the equations into other forms, combining equations in several ways, etc. Does anyone have a sound method for solving this type of problem?

Any help GREATLY appreciated. Thanks!

### RE: needs to solve system of 4 eqns, 4 unknowns

Assuming vali are your 4 unknowns you don't have 4 equations and 4 unknowns. You have 3 equations and 4 unknowns. Which is not analytically solvable. It is a parametric solution.

Fe

### RE: needs to solve system of 4 eqns, 4 unknowns

You need to look at your original derivation of these equations so determine if they are correct. Or if you can formulate another equation.
If you are 100% certain they are correct relations then a numerical solution may be possible. This is a real 'may' though.

Fe

### RE: needs to solve system of 4 eqns, 4 unknowns

(OP)
FeX32,

My bad, I mis-entered the equations in first post

sqrt(x^2+y^2) + t + sqrt((val1-x)^2 + (y-val2)) + Const1 = 0

sqrt(x^2+y^2) + t + Const2 = 0

sqrt(x^2+y^2) + t + 3*sqrt((x-val3)^2 + (z-val3)) + Const3 = 0

sqrt(x^2+y^2) + t + 2*sqrt((val4-y)^2 + (z-val2)) + Const4 = 0

x, y, z, and t are unknown. All other conditions same as before. Also keep in mind these are not my exact equations. val1, val2, val3, val4 are known.

### RE: needs to solve system of 4 eqns, 4 unknowns

So, why not use the numerical solver?

### RE: needs to solve system of 4 eqns, 4 unknowns

once you combine (2) with the remaining three you have 3 eq. 3 unknowns, followed by a bit more leg work on your part, you end up with a quadratic polynomical in y which can be solved easy enough. a bit of back substitution and you are done

### RE: needs to solve system of 4 eqns, 4 unknowns

hacksaw got it, t vanishes and you are left with some ugly eqns

-Const2 + sqrt((val1-x)^2 + (y-val2)) + Const1 = 0

-Const2 + 3*sqrt((x-val3)^2 + (z-val3)) + Const3 = 0

-Const2 + 2*sqrt((val4-y)^2 + (z-val2)) + Const4 = 0

reoorg 1 to give y in terms of x, chuck that into 3, reorg that to give x in terms of z, chuck that into 2 for a nasty polynomial in z.

Cheers

Greg Locock

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### RE: needs to solve system of 4 eqns, 4 unknowns

gL

sqrt((val1-x)^2 + (y-val2)) = Const1'

sqrt((x-val3)^2 + (z-val3)) = Const3'

sqrt((val4-y)^2 + (z-val2)) = Const4'

square all eq

(val1-x)^2 + (y-val2) = Const1"

(x-val3)^2 + (z-val3) = Const3"

(val4-y)^2 + (z-val2) = Const4"

eliminate z terms from (2) & (3)

now two eqn left so sub for x terms in (1)
plot eqn left standing against y and back sub the root(s)
for x & z

rainy day fun for op

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