Thread: linear equations with three unknown variables

I have three equations with three unknown variables. The form of the equation is:

c1*x + c2*y + c3*z = 0
c4*x + c5*y + c6*z = 0
c7*x + c8*y + c9*z = 0

All three equations are equal to zero. The constants are known.
Is there a simple and general way to find the values for x,y and z without all the variables being zero?

Thank you.

Originally Posted by matthijs01

I have three equations with three unknown variables. The form of the equation is:

c1*x + c2*y + c3*z = 0
c4*x + c5*y + c6*z = 0
c7*x + c8*y + c9*z = 0

All three equations are equal to zero. The constants are known.
Is there a simple and general way to find the values for x,y and z without all the variables being zero?

Thank you.

Do you know anything about matrices? If so you can view this system of equations (called homogenous because we set them all equal to 0) as a matrix.

c1 c2 c3
c4 c5 c6
c7 c8 c9

It is a fact that these equations have nontrivial solution when set equal to 0 (the trivial solution is if you set x=y=z=0) if and only if the determinant of the matrix is 0. If the determinant is 0 there is a way to come up with a general formula for solutions. Look up Gauss-Jordan elimination.

The simple way to solve it is:

1. Define z from the first eq. as z = -(c1*x+c2*y)/c3
2. Replace z in 2d and 3rd eq.
3. Solve the system of 2 eq.
4. You will have solution for x and y and you can fine z then.