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$$ \begin{pmatrix} 0 & 1 & -1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ V & -a & 0 & 0 & -d & 0 & 0 \\ 0 & 0 & -b & 0 & d & -e & 0 \\ 0 & 0 & 0 & -c & 0 & e & -f \end{pmatrix} \mathbf{x} = \mathbf{0} $$

Where $$\mathbf{x} = \begin{pmatrix} 1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \\ x_7 \end{pmatrix}$$ How would one go about solving for $f(V,a,b,c,d,e)$? Is Gauss-Jordan elimination the right way to go?

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    Is x a known vector? If you know what x is then the dot product of the last row with x can easily be solved for f. If you don't know what x is then f will also be a function of x and is unlikely to be unique. – 2017-02-07
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    The first element of $\mathbf{x}$ is 1, the rest are variables. – 2017-02-07

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