Why Can't a Linear Equation Have Two Solutions?

A linear equation in one variable cannot have two distinct solutions. Let's explore why.

Consider the linear equation:

$$ ax = b $$

where $ a $ and $ b $ are numbers and $ x $ is the unknown variable.

The coefficients $ a $ and $ b $ can either be zero or non-zero.

Let's analyze each case individually.

1] Case $ a \ne 0 $

If $ a $ is not zero, we can solve the equation by isolating $ x $:

$$ x = \frac{b}{a} $$

In this scenario, the equation has a unique solution, given by $ x = \frac{b}{a} $.

2] Case $ a = 0 $

In this situation, we need to consider two sub-cases:

$ b \ne 0 $
If $ a = 0 $ and $ b \ne 0 $, the equation becomes $$ 0 \cdot x = b $$. This is a contradiction because 0 cannot equal a non-zero number. Therefore, there are no solutions.
$ b = 0 $
If $ a = 0 $ and $ b = 0 $, the equation becomes: $$ 0 \cdot x = 0 $$. This is an identity true for any value of $ x $. In this case, the equation has infinite solutions.

Conclusion

In conclusion, from our analysis, we can state that:

Therefore, a linear equation in one variable can never have exactly two distinct solutions.