\( 3(x-2)=-4(4+3 x)+15 x \) Conditional Identity

To solve the equation \( 3(x-2)=-4(4+3 x)+15 x \), we first simplify both sides. Distributing gives us \( 3x - 6 = -16 - 12x + 15x \). Combining like terms on the right results in \( 3x - 6 = -16 + 3x \). Now let's isolate the variable! Subtract \( 3x \) from both sides to find \( -6 = -16 \), which is not true. Since the equation leads to a contradiction, it indicates that there are no solutions, revealing something interesting about the relationship between \( x \) and the constants provided. This equation perfectly showcases how understanding the structure of equations can save time and effort in solving while also emphasizing the importance of checking solutions. If you find yourself with a statement that doesn't hold, it’s a sign that the variables have no common ground—a classic case of parallel lines never meeting!