GATE Electronics and Communications (EC) 2020 Solved Papers

Question : 44 of 65

Marks: +1, -0

Consider the following system of linear equation

x_{1}+2 x_{2}=b_{1} ; 2 x_{1}+4 x_{2}=b_{2}

3 x_{1}+7 x_{2}=b_{3} ; 3 x_{1}+9 x_{2}=b_{4}

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

Solution:

\table X_1+2X_2=b_1;2X_1+4X_2=b_2\}2b_1=b_2

3 X_{1}+7 X_{2}=b_{3} . . .

(i)

3 X_{1}+9 X_{2}=b_{4} . . .

(ii)
In eqn. (i) we can write as

3 {X}_{1}+6 {X}_{2}+ {X}_{2}= {b}_{3}

3 {b}_{1}+ {X}_{2}= {b}_{3}

{X}_{2}= {b}_{3}-3 {b}_{1}

and in eqn. (ii)

3 X_{1}+6X_{2}+3 X_{2}=b_{4}

3 b_{1}+3[b_{3}-3 b_{1})=b_{4}

-6 b_{1}+3 b_{3}-b_{4}=0

6 b_{1}-3 b_{3}+b_{4}=0

option (B) is correct

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