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Solving a system of linear equations can be approached with different methods, one of which is the substitution method. This technique involves expressing one variable in terms of another from one equation and then substituting this expression into the other equations. This method helps in reducing the number of variables and equations step by step until you can solve for the unknowns.

Let's clarify this method with an example from the exercise given. Initially, equation 1 is manipulated to express variable A in terms of B, as shown in the solution. By substituting this expression into the other equations, we effectively eliminate A from those equations. The result is a simpler set of equations that can be solved sequentially to find the values of B and C, and eventually A. It is essential to execute the substitution correctly and maintain the balance of the equation throughout the process.

Simplifying equations is a fundamental step when solving systems of linear equations. It involves breaking down complex expressions into simpler, more manageable components. This usually includes combining like terms, removing parentheses, and dividing or multiplying both sides of the equation by a common factor to reduce the coefficients.

In our exercise, the second equation is simplified by dividing all terms by 2. This step makes the equation easier to manipulate in later stages. Simplification helps to see the structure of the equations more clearly and can often reveal a direct path to the solution. When simplifying, it's crucial to do the same operation to both sides of the equation to maintain equality. Moreover, ensuring the simplified form is completely reduced primes the equation for efficient substitution or elimination.

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding how to manipulate these expressions is instrumental in solving algebra problems, including systems of equations. In our example, algebraic expressions are simplified and rearranged to isolate variables, enabling us to utilize the substitution method effectively.

For instance, when we express A as 3 - B, we are crafting a new algebraic expression that can be used to replace A in other equations. Recognizing how terms in an expression relate and interact with one another is crucial, such as knowing that a term with A can be substituted with its equivalent expression from another equation. Mastering the manipulation of algebraic expressions not only aids in solving equations, but also in understanding the relationships between mathematical quantities.