# Consider the following system of equations: $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y-7z=c$ where a,b and c are real constants. Then the system of equations : Option: 1 has no solution for all a,b and c. Option: 2 has a unque solution when $5a = 2b+c$ Option: 3 has unique solution for all a,b and c Option: 4 has infinite number of solutions when $5a = 2b+c$

\begin{aligned} &P_{1}: x+2 y-3 z=a\\ &P_{2}: 2 x+6 y-11 z=b\\ &P_{3}: x-2 y+7 z=c\\ &\text { Clearly }\\ &5 \mathrm{P}_{1}=2 \mathrm{P}_{2}+\mathrm{P}_{3} \quad \text { if } 5 \mathrm{a}=2 \mathrm{~b}+\mathrm{c} \end{aligned}

⇒ All the planes sharing a line of intersection

⇒ infinite solutions

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