Simplify the expression:
$
\begin{array}{l}
3a + 3 + 2a + 1
\end{array}
$
The task is to simplify the algebraic expression \(3a + 3 + 2a + 1\). This expression consists of variable terms (containing the variable \(a\)) and constant terms (numerical values). To simplify it, we need to group the like terms together. Like terms are those that have the same variable raised to the same power. In this expression, \(3a\) and \(2a\) are like terms, and \(3\) and \(1\) are like terms.
To simplify the expression, we use the commutative and associative properties of addition, which allow us to rearrange the terms and group them as follows:
\[(3a + 2a) + (3 + 1)\]
Grouping these allows us to add the coefficients of \(a\) and the constants separately.
Now, we add the variable terms together and the constant terms together.
For the variable terms:
\[3a + 2a = (3 + 2)a = 5a\]
For the constant terms:
\[3 + 1 = 4\]
Combining these two results, we get the simplified expression:
\[5a + 4\]
After combining the like terms, the simplified form of the original expression \(3a + 3 + 2a + 1\) is \(5a + 4\). This represents the most compact form of the given expression.
The final result is \(\boxed{5a + 4}\).
Simplifying algebraic expressions is a foundational skill used in many real-life applications, such as budget management or inventory tracking. For instance, if you are calculating the total cost of purchasing multiple items with a base fee, such as three packs of stationery costing \(\$a\) each, a \(\$3\) delivery fee, two more packs costing \(\$a\) each, and an additional \(\$1\) processing fee, your total cost can be represented as \(3a + 3 + 2a + 1\). Simplifying this to \(5a + 4\) allows you to quickly calculate the total cost for any price \(\$a\) per pack of stationery without having to perform multiple additions every time.