Question

An equation of the form $px+qy+r=0$ is known as a linear equation in two variables, where x and y are variables and p, q and r are real numbers. Then, which of the following can never be true?

• A. $p=0$ and $q=0$
• B. $p^2+q^2\ne 0$
• C. $p^2-q^2\ne 0$
• D. $p\ne 0$ and $q\ne 0$

Answer 1

The correct option is A $p=0$ and $q=0$

In a linear equation $px+qy+r=0$ in two variables x and y, the coefficient of both the variables must not be equal to zero simultaneously.
$\Rightarrow p\ne 0$ and $q\ne 0$
The above condition is also represented as $p^2+q^2\ne 0$.
The condition $p^2–q^2\ne 0$ represents that $(p–q)$ and $(p+q)$ both must not be zero. This is also a possible case where $p\ne \pm q$.
Thus, only the condition $p=0$ and $q=0$ is not possible for the given equation $px+qy+r=0$ to be a linear equation in two variables.
Hence, the correct answer is option (a).