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If the system of equations  \begin{align*} 6x-4y&=a,\\ 6y-9x &=b. \end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero, find $\frac{a}{b},$ assuming $b$ is nonzero.

Guest Oct 22, 2017
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6x - 4y  = a        →  6x - 4y  =  a  →  6x - 4y  =  a

6y - 9x  = b        → -9x + 6y =  b  →  6x - 4y  = (-2/3)b

The determinant of the coefficient matrix  = 0

If a = (-2/3)b, we have infinite solutions where x, y are both non-zero

So... we have infinite solutions when   a/b =  -2/3

CPhill  Oct 23, 2017

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