To determine which of these equations is not a linear equation in two variables, let's examine each option:

1. $2x = 3$:

   • This is an equation in one variable (only $x$ is present), so it is not a linear equation in two variables.

2. $x^2 + x = 1$:

   • This equation contains $x^2$, which makes it nonlinear. Linear equations must only have variables to the power of 1. Thus, this is not a linear equation.

3. $4 = 5x - 4y$:

   • This is a linear equation in two variables, $x$ and $y$, with both variables to the power of 1.

4. $x - 2y = 3$:

   • This is also a linear equation in two variables, $x$ and $y$, with both variables to the power of 1.

Conclusion

The equations that are not linear equations in two variables are:

• $2x = 3$
• $x^2 + x = 1$

The best answer based on the options provided is (ii) $x^2 + x = 1$, as this is the only equation with a nonlinear term.

Would you like further details on why certain equations qualify as linear?

Here are five additional questions to deepen understanding:

1. What characteristics define a linear equation in one variable?
2. Why is the presence of an $x^2$ term significant in identifying nonlinearity?
3. How can we identify the number of variables in an equation?
4. Why is the equation $4 = 5x - 4y$ considered linear despite the negative sign?
5. How do equations in one variable differ in solution methods from those in two variables?

Tip: Linear equations in two variables always appear in the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ have no exponents other than 1.