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The substitution method is a powerful tool for solving algebraic equations and verifying solutions. It involves replacing a variable with a given number to determine if the equation is satisfied. To properly utilize substitution, students are advised to follow specific steps.

Begin by identifying the variable within the equation, in this case, it would be 'd'. Then, take the number you wish to check, here it's 6, and substitute it for the variable. This transforms the equation into a series of arithmetic operations using known numbers. Effectively, substitution converts an algebraic equation into a simpler numerical one which can be easily evaluated. After simplification, if both sides of the equation are equal, the number you substituted is indeed a solution to the equation.

Simplifying expressions is a fundamental part of solving algebraic equations. It involves combining like terms and performing arithmetic operations to reduce an expression to its simplest form. In the equation from our exercise, simplification occurs after the substitution of the number 6 for the variable 'd'.

Here's a breakdown: the left-hand side changes from \(3 + 2d\) to \(3 + 2 \times 6\), which simplifies to \(3 + 12\), and then to \(15\). Similarly, the right-hand side, initially \(9 + d\), becomes \(9 + 6\), simplifying to \(15\). When simplifying, it's essential to keep the equation balanced by performing the same operations on both sides. Only when both sides are simplified to the most basic arithmetic expression can you effectively compare and verify the possible solution.

Equation solutions are the values that satisfy the given equation, making it a true statement. When both sides of the equation match after we've substituted a number for a variable and simplified the expressions, it confirms that we've found a solution.

In reference to our exercise, the number 6 was presented as a possible solution. Through substitution and simplification, both sides of \(3 + 2d = 9 + d\) equaled 15 when \(d = 6\), validating that 6 is indeed a solution to the equation. Identifying and verifying solutions is an essential skill in algebra as it allows students to understand the relationships between variables and constants within mathematical contexts.