Question

If the system of linear equations
$x_1+2x_2+3x_3=6$
$x_1+3x_2+5x_3=9$
$2x_1+5x_2+a{x}_3=b$
is consistent and has infinite number of solutions, then?

• A. $a\in R-\left\{8\right\}$ and $b\in R=\left\{15\right\}$
• B. $a=8$, b can be any real number
• C. $a=8,b=15$
• D. $b=15$, a can be any real number

Answer 1

The correct option is C $a=8,b=15$

For consistent and has infinite Number of solution, we have to create linear equations such that it is less then Number of unknown.

$\therefore$ now, $x_1+2x_2+3x_3=6-------\left(1\right)$

$x_1+3x_2+5x_3=9--------\left(2\right)$

$2x_1+5x_2+a{x}_3=b-------\left(3\right)$

now, here adding $\left(1\right)$ & $\left(2\right)$ get equation $\left(3\right)$, so,

$2x_1+5x_2+8x_3=15----------\left(4\right)$

now, compare equation $\left(3\right)$ & $\left(4\right)$.

$a=8$ & $b=15$

For Consistent and Infinite Number of solution $a=8$ & $b=15$.