Question

Consider the system of linear equations ;
$x_1+2x_2+x_3=3$
$2x_1+3x_2+x_3=3$
$3x_1+5x_2+2x_3=1$
The system has

• A. exactly $3$ solutions
• B. a unique solution
• C. no solution
• D. infinite number of solutions

Answer 1

The correct option is C no solution

$x_1+2x_2+x_3=3$ ......$\left(1\right)$

$2x_1+3x_2+x_3=3$ ......$\left(2\right)$

$3x_1+5x_2+2x_3=1$ ......$\left(3\right)$

Adding equations $\left(1\right)$ and $\left(2\right)$

We have $(x_1+2x_2+x_3)+(2x_1+3x_2+x_3)=3+3$

$\Rightarrow 3x_1+5x_2+2x_3=6$

But this contradicts the third equation, i.e., $3x_1+5x_2+2x_3=1$

As such the system is inconsistent and hence it has no solution.