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Question on reducing 3 equations with 3 unknowns

  1. Jul 26, 2012 #1
    Say I have a system of equations of the following form:

    a_1 A^2 + b_1 B^2 + c_1 C^2 + d_1 = f_1
    a_2 A^2 + b_2 B^2 + c_2 C^2 + d_2 = f_2
    a_3 A^2 + b_3 B^2 + c_3 C^2 + d_3 = f_3

    Where a,b,c,d,f are coefficients, and A,B and C are unknown variables.

    1. Can we write them in terms of two variables only?
    2. Can we write them in terms of one variable only?
     
  2. jcsd
  3. Jul 26, 2012 #2

    jedishrfu

    Staff: Mentor

    If you replace the A^2 with AA and similarly with B and C then you have 3 linear eqns in AA, BB, and CC that can be easily solved using matrix methods.
     
  4. Jul 26, 2012 #3
    I was asking because I was reading a document where the author deals with 3 equations and 3 unknown parameters, each equation has quadratic dependence on each of the 3 parameters. Then the author says: with these three equations and three parameters we can reduce the equations to two second order equations with respect to two unknowns (call this case 1), or to one quartic equation with respect to one unknown (call this case 2). So I wanted to know how to do this. I'll do the exercise by eliminating one of the variable and see what I get.
     
  5. Jul 27, 2012 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Ad jedishrfu said, in the given equation, since only the squares of the unknown values occur, we can treat them as, say X=A^2, Y= B^2, and Z= C^2. Then the equations become a_1X+ b_1Y+ c_1Z= f_1, a_2X+ b_2Y+ c_2Z= f_2, and a_3X+ b_3Y+ c_3Z= f_3. (There is no need for the separate 'd_1' and 'f_1'- we can always subtract d_1 from both sides of the equation.)

    There are many ways to solve a "system of equations" but, yes, they all basically involve reducing from three equations in three unknowns to two equations in two unknowns and then to one equation in one unknown. For example, if a_2 is non-zero, I can solve the first equation for X: a_1X= f_1- b_1Y- c_1Z so X= (f_1-b_1X- c_1Z)/a_1. Now replace "X" in the other two equations by that to get two equations in Y and Z only.

    Or:
    1) multiply the first equation by a_2 to get
    a_1a_2X+ b_1a_2Y+ c_1a_2Z= f_1a_2 and
    2) multiply the second equation by a_1 to get
    a)a_1a_2X+ a_1b_2Y+ a_2c_3Z= a_2f_3. Now
    3) subtract those two equation. Since the X term in each equation has the same coefficient, a_1a_2, they cancel leaving a single equation in Y and Z.

    Do the same with, say, the first and third equation, to get a second equation in Y and Z.
     
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