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The condition that the equation $ax + by + c = 0$ represent a linear equation in two variables is

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a \neq 0, b = 0

b \neq 0, a = 0

a = 0, b = 0

a \neq 0, b \neq 0

answer is D.

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Detailed Solution

A linear equation in two variables, x and y, is of the following type.

ax + by + c = 0

, where a, b and c are real numbers.
If either

a = 0

b = 0

then, the equation becomes a linear equation in one variable.

ax + (0) y + c = 0

⇒

ax + c = 0

, which is a linear equation in one variable.
If both a and b are zero then, there is only a constant term left in the equation.

(0) x + (0) y + c = 0

⇒

c = 0

, only constant term left i.e., zero.
Therefore, the condition that makes an equation a linear equation in two variables (x and y) is

a \neq 0, b \neq 0

.
Hence, the correct option is 4.

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