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The substitution method is a fundamental technique in algebra used to find the value of a variable that satisfies a given equation. It involves replacing the variable with a specific value and then solving the equation. When you are given an equation like the one in the exercise, \(5a - 3 = 2a + 6\), and a number to check, such as 3, you substitute that number for the variable 'a'.

Let's dive deeper into the mechanics of substitution. Imagine you are holding a place for your friend 'a' in a line. When your friend arrives, you let them take your spot—this is what substitution is about. In algebraic terms, wherever you see an 'a' in the equation, you replace it with 3 (since that's the number given in this problem). After substitution, your only task is to simplify and verify if both sides of the equation balance out. This approach is not just simple; it's also highly effective when it comes to solving and checking the validity of the possible solutions.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In our example, \(5a - 3\) and \(2a + 6\) are algebraic expressions. Think of these expressions like puzzles. Each piece (number, variable, and symbol) has its place that contributes to the complete picture (the expression's value).

To handle these expressions, it's important to understand how they are structured. In our expressions, '5a' means '5 times a', and similarly, '2a' means '2 times a'. The subtraction and addition are operations that tell you what to do with the numbers and variables. By mastering the art of manipulating these expressions—combining like terms and applying arithmetic operations—you unlock the ability to transform and solve even more complex algebraic challenges.

Equation simplification is the process of making an algebraic equation easier to handle by performing arithmetic operations and combining like terms. In our exercise, simplifying the equation \(5a - 3 = 2a + 6\) involves a few basic steps. After the substitution step, we simplify both sides of the equation to their lowest terms.

Let's break it down: On one side we have \(5*3 - 3\), which reduces to \(15 - 3 = 12\). On the other, we have \(2*3 + 6\), simplifying to \(6 + 6 = 12\). The aim of simplification is to make the equation clearer and the arithmetic manageable, bringing you one step closer to determining the validity of the solution. This step is like cleaning up your room so you can effortlessly see if your keys (in our case, the solution) are right there on the dresser.