\( a x - a - b x + b = 0 , a \neq b \)

To solve the equation \( a x - a - b x + b = 0 \), we can rearrange it. Combine like terms: \( (a - b)x + (b - a) = 0 \). Factor out the constants: \( (a - b)(x - 1) = 0 \). Since \( a \neq b \), we can safely divide both sides by \( a - b \), leading to \( x - 1 = 0 \) and thus \( x = 1 \). Now, let's have some fun with this situation! Did you know that solving linear equations like this one is essential in various fields, from engineering design to finance? Imagine needing to calculate how to balance costs and resources for a project. A simple equation can save professionals hours of brainstorming and lead them to brilliant solutions! Common mistakes arise when variables and constants get mixed up in algebraic manipulations. One common blunder is forgetting to distribute correctly when combining terms or misplacing negative signs! Always double-check your steps; catching those little errors can make a big difference in finding the correct solution. Keep practicing, and you'll become an equation-solving wizard in no time!