The condition for the equation ax + by + c = 0 to represent a linear equation in two variables:

A linear equation in two variables represents a straight line on a Cartesian plane. The equation ax + by + c = 0 is in the standard form of a linear equation, where 'a', 'b', and 'c' are constants, and 'x' and 'y' are the variables.

Key Points:
- In order for the equation to represent a linear equation in two variables, the coefficients 'a' and 'b' must not both be zero.
- If either 'a' or 'b' is zero, the equation will not represent a linear equation.

Explanation:

1. Non-zero Coefficients:
To represent a linear equation, the coefficients 'a' and 'b' in the equation ax + by + c = 0 must not both be zero. If both 'a' and 'b' are zero, the equation becomes 0x + 0y + c = 0, which simplifies to c = 0. This equation represents a constant value rather than a linear relationship between 'x' and 'y'.

2. Constant 'c':
The constant 'c' in the equation ax + by + c = 0 does not affect whether the equation is linear or not. It represents the intercept of the line on the y-axis when x = 0 or on the x-axis when y = 0. However, the value of 'c' is not necessary for determining if the equation is linear or not.

3. Example:
Let's consider a few examples to understand the condition for a linear equation in two variables:

- 2x + 3y - 4 = 0: This equation has non-zero coefficients '2' and '3', satisfying the condition for a linear equation.

- 0x + 5y - 7 = 0: Although the coefficient 'a' is zero, the equation is still linear because 'b' is non-zero.

- 0x + 0y + 6 = 0: This equation has both 'a' and 'b' as zero, violating the condition for a linear equation. It represents a constant value.

Conclusion:
In summary, for the equation ax + by + c = 0 to represent a linear equation in two variables, the coefficients 'a' and 'b' must not both be zero. The constant 'c' does not affect the linearity of the equation. By following this condition, we can determine whether an equation is linear or not and represent it as a straight line on a Cartesian plane.

a , b ,x and y shud not be zero