Question on reducing 3 equations with 3 unknowns

[majon]

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?

 

[jedishrfu]

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.

 

[majon]

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.

 

[HallsofIvy]

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.

 

[blue_raver22]

1. Yes, it is possible to reduce these equations to two variables. This can be done by using substitution or elimination methods. By solving for one variable in terms of the other two in one equation, we can then substitute this expression into the other two equations, effectively reducing the system to two equations with two unknowns.

2. It is not possible to write these equations in terms of one variable only. This is because each equation contains three unknown variables, and in order to solve for one variable, we need to have at least three equations. Therefore, we cannot reduce the system to a single equation with a single unknown variable.