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The equations.

• $x_{1}+2x_{2}+3x_{3}=1$
• $x_{1}+4x_{2}+9x_{3}=1$
• $x_{1}+8x_{2}+27x_{3}=1$

have

1. Only one solution.
2. Two solutions.
3. Infinitely many solutions.
4. No solutions

This is non homogeneous equation ...for such type of questions we have to check following 3 cases :

1)if rank(A)<rank(A|B) then AX=B has no solution

2)if rank(A|B)=rank(A)=no of unknowns then AX=B has unique non-zero solution

3)rank(A|B)=rank(A)<no of unknowns then infinite no of solution

here I'm getting rank(A|B)=rank(A)=no of unknowns =3 so only one solution..