When does a linear equation have infinitely many solutions?

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For which values of $k$ the equation $(k+8)x-3k = 3k(1-x)-6$ has infinitely many solutions?

Since this is not a set of linear equations you can't use the augmented matrix and Gaussian elimination, how would you solve this?

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1 Answer

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The equation with variable $x$ is given by$$ x(4k + 8) - 6(k + 1)=0. $$For $k=-2$ it has no solution over a field of characteristic zero, hence we may divide by $4k+8$ to obtain a unique solution for $x$. In particular, the equation never has infinitely many solutions.

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