# Homogeneous system

1. Apr 15, 2013

### scientifico

Hi,
I'm solving out a homogeneous 3 equations and 4 variables system so I considered one variable as known term but the determinant of the matrix is 0, how do I use Cramer in this case ?
these are the 3 equations

2x + 3y - z - 2v = 0
4x - 3y - 5z + 5v = 0
8x + 3y - 7z + v = 0

determinant of {{2,3,-1},{4,-3,-5},{8,3,-7}} is 0

thanks

2. Apr 15, 2013

### LCKurtz

The normal way to solve a system like this would be to use row reduction.

3. Apr 15, 2013

### scientifico

but how can I solve it with Cramer ?

4. Apr 15, 2013

### Staff: Mentor

I don't think there is any reason that you should assume that one variable is known. This is apparently a system of three equations in four unknowns. The matrix of coefficients isn't square, so the concept of the determinant doesn't apply, and you can't use Cramer's Rule.

5. Apr 15, 2013

### Staff: Mentor

6. Apr 15, 2013

### LCKurtz

Also note that it isn't obvious just looking at the system whether there might be 1 or more free variables. You really need row reduction to answer that.

7. Apr 16, 2013

### Ray Vickson

Cramer's rule only applies when the determinant of coefficients is nonzero. No matter which column you omit, the resulting 3x3 matrix has zero determinant, so in this question Cramer's Rule fails in every single case you try.

As others have suggested, just use row reduction; but an equivalent description would be: just use the variable-elimination method as taught in high school.

Last edited: Apr 16, 2013