That will be dependent system. You'll have parametric solution.

sorry, can you simplify your answer, I am not a mathematician

can we use Matlab for example to solve this

parametric means solution in terms of another variable... for example: x+y=0 and x+z =1 => y=-x and z=1-x. The solution is in terms of x. If you give value for x you can give value for y and z (e.g., x=0 so y=0 and z=1). that's why they are dependent.

If you dont want this then have 3 unknowns and 3 equations (it is possible, but not always, to have non parametric solution).

Try to find Internet sources on something like "system of (linear) equations". In Hong Kong, it is a topic in the secondary school mathematics curriculum.

As far as I understood, my problem is not a system of linear equations

Your example: x+y=0 and x+z =1 is a system of linear equations. If you have nonlinear equations, you can treat your problem as getting intersections of curves or surfaces. Matlab have functions for this. But I'm already giving you a caution that looking for solutions are difficult, especially if you have 3 or more equations.

if you have polynomial system: try to look at this http://math.berkeley.edu/~bernd/cbms.pdf

Dear Ahmed,

As soon as you have a system of two equations in three variables, you would have an infinite number of solutions. Therefore, you can of course solve the system of equations, although there will be no unique solution. For example, in your case, you have x + y = 0 and x + z = 1. Accordingly, you have to fix one of the three variables, and solve for the other two. For example, x = 0 would give y = 0 and z = 1.

x can take an uncountably infinite number of values, and therefore for every x, you would ultimately get an uncountably infinite number of values of y and z.

This is certainly true for some LINEAR systems only:

> As soon as you have a system of two equations in three variables, you would have >an infinite number of solutions.

x+y=1

x-y=1

2x+2y=2

This system has a unique solution.

The nonlinear equation in three variables

x^2+y^2+z^2=0

has a unique solution.

Dear Janos,

1) The example cited in the question was discussed in my answer. I did not cite an example on my own.

2) The system x + y = 1, x - y = 1, 2x + 2y =2, is actually nothing but the system x + y = 1 and x - y = 1, which is of course solvable uniquely. How is it related to the question? The question was regarding two equations in three variables. Where is third variable here?

3) x^2 + y^2 + z^2 = 0 has a unique solution because only (0, 0, 0) satisfies this equation. How about x^2 + y^2 + z^2 = 1 for example? Would you still get a unique solution? In x^2 + y^2 + z^2 = 0, we have actually utilized the fact that when a sum of squares is zero, the variables are not actually variables but constants equal to zero. How about x^3 + y^3 z^3 = 0, for example?

You can solve the system of equations, although there will be no unique solution. For example, in your case, you have x + y = 0 and x + z = 1. Accordingly, you have to fix one of the three variables, and solve for the other two. For example, z = 0 would give x= 1 and y =- 1.

There is a web page where they solve the system that you have mentioned. Before that, you have to rewrite the linear system in its complete form, like this:

1* x + 1 * y + 0 * z = 0 ------- (1)

1* x + 0 * y + 1 * z = 1 --------(2)

and work with the extended matrix.

The web page address (URL) is:

http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi

good luck!

In this system,

Number of unknowns (n=3) less than number of equations (k=2), so it has infinitely many solutions which can be expressed by arbitrary parameters, we have (n-k) free parameters, i. e. (3-2=1) one free parameter say t

The solution will be:

x = t

x+y =0 i.e: y = - x

y = - t

x+z =1 i.e z = 1 - x

z = 1 - t

Once you take any value for the parameter t, you will get x, y and z values which satisfies the two equations.

For example:

for t=1

x = 1, y = -1 and z = 0

Best Luck.

I would recommend Equation Solving module at Wolfram|Alpha:

https://www.wolframalpha.com/examples/mathematics/algebra/equation-solving/