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GATE 2020 EC – Question 36

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Consider the following system of linear equations.

$$x_1+2x_2=b_1;\quad2x_1+4x_2=b_2;\quad3x_1+7x_2=b_3;\quad3x_1+9x_2=b_4$$

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

  1. $b_2=2b_1$ and $6b_1-3b_3+b_4=0$
  2. $b_3=2b_1$ and $6b_1-3b_3+b_4=0$
  3. $b_2=2b_1$ and $3b_1-6b_3+b_4=0$
  4. $b_3=2b_1$ and $3b_1-6b_3+b_4=0$

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Correct answer: (A) $b_2=2b_1$ and $6b_1-3b_3+b_4=0$

Explanation

The second equation is twice the first, so $b_2=2b_1$ is needed. Solving the first and third equations gives $x_2=b_3-3b_1$ and $x_1=7b_1-2b_3$. Substituting in the fourth: $3(7b_1-2b_3)+9(b_3-3b_1)=-6b_1+3b_3=b_4$, i.e. $6b_1-3b_3+b_4=0$.