VBA How to solve two equations with two unknowns

I am trying to calculate point on a line. I got the points of the edges and one distance between one edge to the point I want to find (which is B).

A(2,4)

B(x,y)

C(4,32)

The distance between A to B is 5.

How can I 开发者_如何学JAVAcalculate Bx and By? using the following equations:

d = Math.Sqr((Bx-Ax)^2 + (By-Ay)^2)
d = Math.Sqr((Cx-Bx)^2 + (Cy-By)^2)

and than compare the equations above.

Here is the equations with the points placed:

5 = Math.Sqr((Bx-2)^2 + (By-4)^2)
23.0713366 = Math.Sqr((4-Bx)^2 + (32-By)^2)

or

Math.Sqr((Bx-2)^2 + (By-4)^2) - 5 = Math.Sqr((4-Bx)^2 + (32-By)^2) - 23.0713377

How can I solve this using VBA?

Thank you!

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I won't solve your equations above because they are an unnecessarily complex way to state the problem (and the existence of a solution is questionable in the presence of rounding), but all the points on the line A=(Ax,Ay) to C=(Cx,Cy) can be described as B=(Ax,Ay) + t*(Cx-Ax,Cy-Ay) with t between 0 and 1.

The distance between B and A is then given by d=t*Sqrt((Cx-Ax)^2+(Cy-Ay)^2), which you can invert to get the proper t for a given d - t=d/Sqrt((Cx-Ax)^2+(Cy-Ay)^2)

In your case, B(t) = (2,4) + t*(2,28), t=5/Sqrt(2^2+28^2) ~ 0.178 -> B ~ (2,4) + 0.178 * (2,28) ~ (2.356, 8.987).

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VBA has no Symbolic Language capability. To solve this problem, there are different approach :

1. Transform the equations to isolate one of the unknowns, most likely to use substitution, and compute it (I recommend this for your problem.)
2. Transform your functions and derive them to use Newton's methods (don't do this, it's overkill.)
3. Use a "brute force" convergence methods : Fix a min/max for each variable and use bisection methods to find what you want (I don't recommend this because you'll most likely "fall" into a local minimum/maximum in your case.)

So basically, I'd say you go with the first way. It requires 15mins of tinkering with mathematical equations, then you're set to go.