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Consider the following system of linear equations.

$\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$

Which one of the following conditions ensures that a solution exists for the above system?

- $b_{2}=2b_{1}$ and $6b_{1}-3b_{3}+b_{4}=0$
- $b_{3}=2b_{1}$ and $6b_{1}-3b_{3}+b_{4}=0$
- $b_{2}=2b_{1}$ and $3b_{1}-6b_{3}+b_{4}=0$
- $b_{3}=2b_{1}$ and $3b_{1}-6b_{3}+b_{4}=0$