# Linear equations with three unknowns

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Ahmed Shatla on 5 Oct 2022
Answered: Paul on 5 Oct 2022
New to matlab, trying to solve something like this (this is just an exampe)
120x - 55y - 75z = 0
111x - 67y - 21z = 0
34x - 92y - 11z = 0
but it is known that x^2 + y^2 + z^2 = 1
im trying to end up with x and z so i can use this ^ to find y
Thank you

Paul on 5 Oct 2022
Hi Ahmed,
The problem in the Question is ill-defined becasue there are no values of (x,y,z) that satisfy all four equations.
If you have a set of equations that does have a solution, fsolve might be useful for example
A = [120 -55 -75;111 -67 -21]; % first two equations from the Question
p = fsolve(@(p) [A*p(:);sum(p.^2)-1],[1 0 0]) % p = [x y z]
Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.
p = 1×3
0.5345 0.8018 0.2673
For this problem, p is simply the unit-norm basis vector for the null space of A, as it should be
null(A)
ans = 3×1
0.5345 0.8018 0.2673

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