Balancing Chemical Equations with Linear Algebra

[Squall]

Homework Statement

Balance the following Chemical Equation
(x1)KMnO4+(x2)MnSO4+(x3)H2O->(x4)MnO2+(x5)K2SO4+(x6)H2SO4

Homework Equations

Here is the matrix I came up with but it seems to be flawed but I am unable to see what i am doing wrong

1 0 0 0 -2 0 (K)
0 1 0 -1 0 0 (Mn)
4 4 1 -2 -4 -4 (O)
0 1 0 0 -1 -1 (S)
0 0 2 0 0 -2 (H)
Each column corresponds to the amount of each element in each compound.

This is the reduced row echelon form of the matrix
1 0 0 0 0 -1/3
0 1 0 0 0-7/6
0 0 1 0 0 -1
0 0 0 1 0-7/6
0 0 0 0 1 -1/6

I seem to be missing the free variable I need to solve this problem

The Attempt at a Solution

The answer i come up with is
X1=(1/3)
X2=(7/6)
X3=(1)
X4=(7/6)
X5=(1)

and here is the problem for I only get five answers and not the six that i need to balance the equation. Can anyone point me in a direction or say what I am doing wrong any help will be appreciated thank you.

 

[yazooj]

you need to add a column of zeros on the right of your matrix

 

[muntazimabbas]

Balance is made by taking x₆ as a free variable. This variable is free because in the Matrix, x₆ is the position where no pivot element is present, i.e. column of x₆ is not a pivot column. So compute the remaining variables in the form of x₆ and then multiply all the values of x₁ ... x₅ by the LCM of the denominators of the fractions.

In my view the answer will look like (for example if I suppose the obtained Echelon form is correct, i.e. This is the reduced row echelon form of the matrix
1 0 0 0 0 -1/3
0 1 0 0 0-7/6
0 0 1 0 0 -1
0 0 0 1 0-7/6
0 0 0 0 1 -1/6)


X₁=(1/3)t
X₂=(7/6)t
X₃=(1)t
X₄=(7/6)t
X₅=(1/6) t
x₆=t, where t is in Set of Real Numbers


OR

X₁=(2)t
X₂=(7)t
X₃=(6)t
X₄=(7)t
X₅=(1) t
x₆=(6)t, where t is in Set of Real Numbers

 

[Borek]

This is an old thread (how did I miss it back then?). Compare algebraic method of balancing chemical equations.