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Consider following system of equations:

$\begin{bmatrix} 1 &2 &3 &4 \\ 5&6 &7 &8 \\ a&9 &b &10 \\ 6&8 &10 & 13 \end{bmatrix}$$\begin{bmatrix} x1\\ x2\\ x3\\ x4 \end{bmatrix}$=$\begin{bmatrix} 0\\ 0\\ 0\\ 0 \end{bmatrix}$

The locus of all $(a,b)\in\mathbb{R}^{2}$ such that this system has at least two distinct solution for ($x_{1},x_{2},x_{3},x_{4}$) is

  1. a parabola
  2. a straight line
  3. entire $\mathbb{R}^{2}$
  4. a point
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The determinant of the matrix is easily calculated: $4a+4b-72$.  Now there are more than one solution iff that determinant equals zero. Clearly the locus is a straight line!

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