How do i solve 2 equations for 2 unknowns?

I have just started to learn MATLAB. I just need to know how do i code the following to find the unknowns and find the intersection points? I have to solve 8 sets of equations and find the intersecting co-ordinates. Here is the format of the equations which is to be used. Except (X,Y) all other values are known in each system of equation. I need to find (X,Y) in each system and find the intersecting point/points.

VT1= sqrt((X1-X)^2+(Y1-Y)^2) and VT2= sqrt((X2-X)^2+(Y2-Y)^2)

VT2= sqrt((X2-X)^2+(Y2-Y)^2) and VT3= sqrt((X3-X)^2+(Y3-Y)^2)

VT3= sqrt((X3-X)^2+(Y3-Y)^2) and VT4= sqrt((X4-X)^2+(Y4-Y)^2)

VT4= sqrt((X4-X)^2+(Y4-Y)^2) and VT5= sqrt((X5-X)^2+(Y5-Y)^2)

VT5= sqrt((X5-X)^2+(Y5-Y)^2) and VT6= sqrt((X6-X)^2+(Y6-Y)^2)

VT6= sqrt((X6-X)^2+(Y6-Y)^2) and VT7= sqrt((X7-X)^2+(Y7-Y)^2)

VT7= sqrt((X7-X)^2+(Y7-Y)^2) and VT8= sqrt((X8-X)^2+(Y8-Y)^2)

VT8= sqrt((X8-X)^2+(Y8-Y)^2) and VT1= sqrt((X1-X)^2+(Y1-Y)^2)

Answer 1

[Roger Stafford]

For each of the eight pairs of equations you are asking for the two intersections of two circles. Here is a direct solution extracted from Answers #132554, 5 Jun 2014, which does not use iteration.

Let P1 = [x1;y1] and P2 = [x2;y2] be column vectors for the coordinates of the two centers of the circles and let r1 and r2 be their respective radii.

   d2 = sum((P2-P1).^2);
   P0 = (P1+P2)/2+(r1^2-r2^2)/d2/2*(P2-P1);
   t = ((r1+r2)^2-d2)*(d2-(r2-r1)^2);
   if t <= 0
     fprintf('The circles don''t intersect.\n')
   else
     T = sqrt(t)/d2/2*[0 -1;1 0]*(P2-P1);
     Pa = P0 + T; % Pa and Pb are circles' intersection points
     Pb = P0 - T;
   end

The coordinates of vectors 'Pa' and 'Pb' are the two possible solutions to the equations of the two circles.

[Swaroop]

Thank you for your prompt response Mr.Roger. But i am not finding the intersection of two circles. Actually i have a problem where i am locating a point or points within a circle based on certain criteria(relevant to time), and only the intersecting point/points is all that i need. 8 system of equations each giving me the co-ordinates(a point in circle) after solving them and finally the intersection of these co-ordinates is all that i need.

[Roger Stafford]

It does not matter what your actual physical problem is. A pair of equations such as:

   VT1= sqrt((X1-X)^2+(Y1-Y)^2) and VT2= sqrt((X2-X)^2+(Y2-Y)^2)

is mathematically EQUIVALENT to finding the intersections between two circles whose centers are at [X1;Y1] and [X2;Y2] with respective radii of VT1 and VT2. I suggest you run the code I gave you, making the appropriate substitution of values, and check to see if each of your results in Pa and Pb satisfy the above equations.

Actually it would be easiest to convert that algorithm to a function which receives as inputs the four quantities, P1, P2, r1, and r2, and outputs Pa and Pb, and call on it appropriately for each of your eight pairs of equations. I leave that task for you to do.

[Ken]

In Robotics we have forward and inverse Kinematics equations i.e. A*Si = B*phi. Can this approach be used to solve these equations?

[Roger Stafford]

Very likely the answer is ‘no’, but you should accurately describe the quantities A, Si, B, and phi to get a reliable answer, and also you should probably establish a separate question for it in this ‘ånswers’ forum.