Question

If the system of equations
$x+y+z=5x+2y+3z=9x+3y+az=\beta$
has infinitely many solutions, then $\beta -\alpha$ equals :

• A. $8$
• B. $18$
• C. $5$
• D. $21$

Answer 1

The correct option is A $8$

$x+y+z=5$........(i)

$x+2y+3z=9$.......(ii)

$x+3y+az=\beta$........(iii)

If system of equation has infinitely many solution, then one equation can be expressed as linear

combination of other two equations.

Let $2\left(ii\right)=\left(i\right)+\left(iii\right)$

$2x+4y+6z-18=2x+4y+(a+1)z-(\beta +5)$

$a+1=6\Rightarrow a=5$

$\beta +5=18\Rightarrow \beta =13$

Therefore, $\beta -a=13-5=8$