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To understand how to translate words into equations, it's crucial to have a firm grasp on algebraic expressions. An algebraic expression is a mathematical phrase involving numbers, operations, and at least one variable. For instance, in the statement from the exercise 'Six less than a number is fifteen,' the 'number' can be represented as a variable, typically denoted by a letter such as 'x'.

Translating the statement into an algebraic expression means we turn the words into a combination of variables and numbers connected by operations. The phrase 'six less than' indicates subtraction, so we write the operation before the variable. This results in an algebraic expression of the form 'x - 6', which can be used to build an equation.

Linear equations are fundamental in algebra and represent a relationship where each term is either a constant or the product of a constant and a single variable. Solving a linear equation like the one from the exercise, 'x - 6 = 15', involves finding the value of the variable that makes the equation true.

The solution process often requires performing operations that will isolate the variable on one side of the equation. In our example, adding 6 to both sides helps keep the balance of the equation and determines the unknown number. This kind of manipulation is at the core of algebra, as it allows students to solve for unknown variables systematically and confidently.

After finding a solution to an equation, it is just as important to ensure that the solution is correct. Verification involves substituting the solution back into the original equation to check if it satisfies the equation. For the given problem, after solving the equation and determining that 'x = 21', we replace 'x' with 21 in the original equation 'x - 6 = 15'.

If both sides of the equation equal the same number after the substitution, the solution is verified. In our exercise, substituting 21 into the equation indeed gives us '21 - 6 = 15', proving that our solution is accurate. This means the solution is not only numerically correct but also logically consistent with the initial mathematical statement. Verification is a crucial step in solving equations because it confirms the reliability of the answer.