# Thread: Two equations with two unknowns: how to rearrange or solve the equations

1. ## Two equations with two unknowns: how to rearrange or solve the equations

I have to equations:
y = T*Cx*U and x = T*Cy*U where

x and y are scalar values that are known,
Cx and Cy are both 4x4 matrices that are known.
T is a 1x4 vector : T=[1 t t^2 t^3] where t is unknown and
U is a 4x1 vector : U=[1 u u^2 u^3]' where ' means transpose and u is unknown.

So basically I have two equations and two unknowns, t and u. I would like to figure out a way to represent the equation so that
the unknowns are a function of the knowns
so that t = f(x,y,Cx,Cy) and u = f(x,y,Cx,Cy) or T = f(x,y,Cx,Cy) and U = f(x,y,Cx,Cy).

I imagine that getting the final numerical solution might involve getting the roots of a cubic but I don't know how to get that far.

I've tried to figure out a way to substitute one equation into the other to eliminate one of the unknowns (i.e., U or T) but I get stuck because I'm not sure what to do about dividing by a vector (i.e., finding the inverse of a vector?).

2. ## Re: Two equations with two unknowns: how to rearrange or solve the equations

can you post the matrix C ? and the values x,y?

3. ## Re: Two equations with two unknowns: how to rearrange or solve the equations

Originally Posted by jakncoke
can you post the matrix C ? and the values x,y?
I can give you example numbers but they are floating point values that I'm pulling from a program I'm working on so I don't know if they would be helpful. In my program I can go from u,t to x,y but I would like to go the other way so I am trying to invert the equation. For this example, t = 0.3624 and u = 0.4950. t and u are always between 0 and 1. (I am trying to invert a bicubic interpolation fn. )

x = 1022.9;
y = 495.9297;

Cx =

930.8436 12.3171 36.8271 -18.4136
220.5397 -2.7589 -8.5661 4.2831
2.2148 0.3214 0.6427 -0.3214
-0.4175 -0.0662 -0.1324 0.0662

Cy =

400.3912 194.3285 -5.4094 2.7047
0.3092 1.2141 0.4406 -0.2203
-0.0395 -0.1551 -0.0583 0.0291
0.0066 0.0139 0.0013 -0.0006