Question

For what values of a and b, the following simultanesous equations have an infinite number of soultions?

$x+y+z=5;$
$x+3y+3z=9;$
$x+2y+az=b;$

• A. 2, 7
• B. 3, 8
• C. 8, 3
• D. 7, 2

Answer 1

The correct option is A 2, 7

The augumented matrix for the given system of equation is

$[A:B]=\left[\begin{array}{cccc}1& 1& 1& |5\\ 1& 3& 3& |9\\ 1& 2& a& |b\end{array}\right]$
$\begin{array}{c}{R}_2\to R_2-R_1\\ R_3\to R_3-R_1\end{array}$$\sim \left[\begin{array}{cccc}1& 1& 1& |5\\ 0& 2& 2& |4\\ 0& 1& a-1& |b-5\end{array}\right]$
$R_2\to \frac{1}{2}{R}_2\sim \left[\begin{array}{cccc}1& 1& 1& |5\\ 0& 1& 1& |2\\ 0& 1& a-1& |b-5\end{array}\right]$
$R_3\to R_3-R_2\sim \left[\begin{array}{cccc}1& 1& 1& |5\\ 0& 1& 1& |2\\ 0& 0& a-2& |b-7\end{array}\right]$
For infinite number of solutions a - 2=0, b - 7=0
Hence a=2 and b=7