Question

The equation $x+2y+2z=1$ and $2x+4y+4z=9$ have

• A. Only one solution
• B. Only two solutions
• C. Infinite number of solutions
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option

Given set of equations are $x+2y+2z=1$ and $2x+4y+4z=9$

We know that, for a pair of equations of line $a_{1}x+b_{1}y+c_{1}z=d_1$ and $a_{2}x+b_{2}y+c_{2}z=d_2$

1. There is a unique solution if,
   $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$ or $\frac{c_1}{c_2}\ne \frac{b_1}{b_2}$ or $\frac{a_1}{a_2}\ne \frac{c_1}{c_2}$
2. There are infinite solutions if,
   $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{d_1}{d_2}$
3. No solution if,
   $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\ne \frac{d_1}{d_2}$

Here, $a_1=1$, $b_1=2$, $c_1=2$, $d_1=1$, $a_2=2$, $b_2=4$, $c_2=4$ and $d_2=9$

We observe that,
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{1}{2}$
And,
$\frac{d_1}{d_2}=\frac{1}{9}$

So, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\ne \frac{d_1}{d_2}$

Therefore, there are no solutions.

Hence, option (D) i.e. None of these is correct.