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Solving a linear equation is like finding the key to a puzzle. Every step you take is a move towards unveiling the number hidden beneath the variable. In the exercise given, the equation is a simple one: 4x - 14 = -82. To solve it, you start by simplifying the equation, which is essentially like tidying up a room to make finding things easier. You move all the variable terms to one side and the constants to the other. This eventually reveals the value of the variable, which in this case is x. Following these steps methodically, ensures you don't lose track of any pieces of the puzzle.

Imagine the variable x as someone you're trying to spotlight on a stage. To isolate x, you need to remove all the other distractions - or in mathematical terms, the other numbers. You do this by doing the opposite operation. If there's subtraction, you add; if there's multiplication, you divide. By adding 14 to both sides of our equation, you move that 'distraction' away and make x the star. Isolating the variable is critical for understanding the relationship between the variable and the rest of the equation.

Remember, what you do to one side, you must do to the other to keep the equation balanced, just like scales. Once the variable is by itself, you are one step closer to finding its value.

To make an expression easier to work with, you simplify it. In our example, after adding 14 to both sides, we need to combine like terms, which is like combining apples with apples, leaving you with a cleaner, simpler expression. This step is where we get 4x = -68 from 4x = -82 + 14. It's important to perform these operations carefully to avoid any mistakes. Simplification is about making the equation as straightforward as possible, so you can easily see the path to the solution.

You've guessed the number of candies in the jar, but how do you know you're right? You count them! Checking the solution in math is similar. Once you think you've solved the equation, you substitute the value back into the original equation to verify it. If both sides equal, you've found the correct solution. In the exercise, substituting x = -17 gives us 4*(-17) - 14 = -82, which holds true upon simplification. This step is like your math victory lap – it confirms your hard work and ensures the variable’s value is indeed the solution to the equation.