Question

Consider the following linear system.

x + 2y - 3z = a

2x + 3y + 3z = b

5x + 9y - 6z = c

This system is consistent if a, b and c satisfy the equation

1. 7a - b - c = 0
2. 3a + b - c = 0
3. 3a - b + c = 0
4. 7a - b + c = 0

Answer 1

Correct Answer - Option 2 : 3a + b - c = 0

Concept:

The system AX = B has

1. A unique solution if and only if Rank of A = Rank of [A|B] = Number of variables
2. Infinitely many solutions if Rank of A = Rank of [A|B] < Number of variables
3. No solution (inconsistent) if Rank of A ≠ Rank of [A|B], i.e Rank of A < Rank of [A|B]

​Calculation:

The system of equations can be written in matrix form as

\(\left[ {\begin{array}{*{20}{c}} 1&2&{ - 3}\\ 2&3&3\\ 5&9&{ - 6} \end{array}\;\left| {\;\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right.} \right]\)

Applying \({R_2} \to {R_2} - 2{R_1}\)

\(\begin{array}{*{20}{l}} {{R_3} \to {R_3} - 3{R_1}}\\ {\left[ {\begin{array}{*{20}{c}} 1&2&{ - 3}\\ 0&{ - 1}&9\\ 0&{ - 1}&9 \end{array}\;\left| {\;\begin{array}{*{20}{c}} a\\ {b - 2a}\\ {c - 5a} \end{array}} \right.} \right]}\\ {{R_3} \to {R_3} - {R_2}}\\ {\left[ {\begin{array}{*{20}{c}} 1&2&{ - 3}\\ 0&{ - 1}&9\\ 0&0&0 \end{array}\;\left| {\;\begin{array}{*{20}{c}} a\\ {b - 2a}\\ {c - 3a - b} \end{array}} \right.} \right]} \end{array}\)

For system to be consistent,

Rank of [A] = Rank of [A|B]

∴  c – 3a – b = 0

It can be written as 3a + b – c = 0