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When solving any mathematical expression, understanding the order of operations is crucial. This concept helps us know which operations to perform first, ensuring we arrive at the correct solution. Think of the order of operations as a recipe for success in arithmetic. In most mathematical communities, the acronym PEMDAS helps students remember this order.

• Parentheses
• Exponents
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

Let's break it down using the expression in the problem: \(3a - (b+c)\). According to the order of operations, the calculation inside the parentheses should be performed first. This means adding \(b\) and \(c\) together. After that, we look at multiplication, which means multiplying 3 by 6. Finally, subtraction is conducted at the end. Always remember to check the order, as doing operations in the wrong sequence can lead to incorrect results.

In expressions involving variables, these variables act as placeholders for numbers. "Variable substitution" is the process of replacing these placeholders with actual numbers. This step is essential before performing any calculations.
Take the original expression: \(3a - (b+c)\). Here, variables \(a\), \(b\), and \(c\) are given specific values: \(a=6\), \(b=10\), and \(c=3\). Variable substitution involves plugging these numbers into respective places in the expression, turning it into a numerical format like this: \(3 \times 6 - (10 + 3)\).
This transformation makes it possible to proceed with arithmetic operations, as it converts an abstract expression into a solvable one. Substitution also helps verify the expression's solution by simplifying and solving step by step.

Arithmetic operations refer to the basic operations we perform on numbers: addition, subtraction, multiplication, and division. In solving expressions, these operations must be performed in a systematic way to get accurate results.
Once we substitute variables with their values, such as turning \(3a - (b+c)\) into \(3 \times 6 - (10 + 3)\), we start by simplifying operations within parentheses first. After parentheses, multiplication and addition often come next.
In our example:

• Calculate the sum inside the parentheses: \(10 + 3 = 13\).
• Next, perform the multiplication: \(3 \times 6 = 18\).
• Finally, perform the subtraction: \(18 - 13 = 5\).

It’s important for students to carefully carry out each step and review their work. Making sure each operation is done in sequence ensures the solution is clean and accurate.