Q72 / 160 · TS EAMCET 2015 · Mathematics
The system of equations $4 x+y+2 z=5$, $x-5 y+3 z=10,9 x-3 y+7 z=20$ has
- Ano solution
- Bunique solution
- Ctwo solutions
- Correct answerDinfinite number of solutions
Solution
We have,
$$
\begin{aligned}
D & =\left|\begin{array}{ccc}
4 & 1 & 2 \\
1 & -5 & 3 \\
9 & -3 & 7
\end{array}\right| \\
& =4(-35+9)-1(7-27)+2(-3+45) \\
& =-104+20+84=0 \\
D_1 & =\left|\begin{array}{ccc}
5 & 1 & 2 \\
10 & -5 & 3 \\
20 & -3 & 7
\end{array}\right| \\
& =5(-35+9)-1(70-60)+2(-30+100) \\
& =-130-10+140=0 \\
D_2 & =\left|\begin{array}{lll}
4 & 5 & 2 \\
1 & 10 & 3 \\
9 & 20 & 7
\end{array}\right| \\
& =4(70-60)-5(7-27)+2(20-90) \\
& =40+100-140=0
\end{aligned}
$$
$$
\begin{aligned}
\text { and } & D_3=\left|\begin{array}{ccc}
4 & 1 & 5 \\
1 & -5 & 10 \\
9 & -3 & 20
\end{array}\right| \\
& =4(-100+30)-1(20-90)+5(-3+45) \\
& =-280+70+210=0
\end{aligned}
$$
So, the given system of equations has infinite number of solutions.
Hence, the correct answer is option: 'D'.