(1) \( 5 x+3=2 x+1 \)

To solve the equation \( 5x + 3 = 2x + 1 \), you want to get all terms containing \( x \) on one side and constant terms on the other. First, subtract \( 2x \) from both sides to get: \( 5x - 2x + 3 = 1 \), which simplifies to \( 3x + 3 = 1 \). Next, subtract 3 from both sides: \( 3x = 1 - 3 \), or \( 3x = -2 \). Now, divide by 3: \( x = -\frac{2}{3} \). Did you know that this type of linear equation is often used to describe things like profit margins or calculating distances? Just like in this equation, real-world problems are often simplified to find solutions quickly and efficiently, whether it’s budgeting finances or determining travel time! When solving algebraic equations, one common mistake is to forget to maintain the balance throughout the process. Always remember to apply operations equally to both sides of the equation. For instance, sometimes students subtract or add a number only to one side, unintentionally leading to incorrect results. Keeping everything balanced ensures you find the correct solution every time!