Problem 25 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$2 a+4+3 a+5$$

Step-by-Step Solution

Answer

The simplified expression is \(5a + 9\).

Step 1: Identify Like Terms

First, look for terms that are alike in the expression \(2a + 4 + 3a + 5\). Like terms are terms that contain the same variable raised to the same power. Here, \(2a\) and \(3a\) are like terms because they both have the variable 'a'. The numbers 4 and 5 are constants and can also be combined.

Step 2: Rearrange Terms Using the Commutative Property

The commutative property of addition allows us to rearrange the terms. We rewrite the expression as \(2a + 3a + 4 + 5\). This makes it easier to combine the like terms.

Step 3: Combine Like Terms

Now, add the coefficients of the like terms. For the 'a' terms, add \(2a\) and \(3a\), which gives \(5a\). For the constant terms, add 4 and 5, which gives 9.

Step 4: Write the Simplified Expression

Combine the results from the previous step into a single expression: \(5a + 9\). This is the simplified form of the original expression.

Key Concepts

Like Terms

When simplifying algebraic expressions, one of the first concepts you will encounter is "like terms." Like terms are terms that have the same variables raised to the same powers. Identifying like terms allows you to combine them accurately.For example: - In the expression \(2a + 4 + 3a + 5\), the terms \(2a\) and \(3a\) are like terms because they both contain the variable 'a'.- Similarly, both 4 and 5 are constants (numbers without variables), so they too can be combined.Recognizing like terms is a critical first step in simplifying expressions as it sets the stage for further simplification. To identify like terms, look for terms that match in variable and exponent.

Combining Like Terms

Combining like terms is a key step in simplifying expressions. After identifying like terms, the next logical step is to combine them.### How to Combine Like TermsHere are the basic steps:

• Look for terms in the expression that have the same variable raised to the same power.
• Add or subtract their coefficients. Coefficients are the numbers in front of the variables.
• Write the result with the common variable.

In the expression \(2a + 4 + 3a + 5\):- The like terms \(2a\) and \(3a\) are combined to become \(5a\) because \(2 + 3 = 5\).- The constants 4 and 5 add up to 9.After combining, the expression simplifies to \(5a + 9\). This process makes the expression simpler and more manageable.

Commutative Property

The commutative property is a foundational principle in mathematics, particularly in simplifying algebraic expressions. It states that numbers can be added or multiplied in any order without changing the result. This property can be incredibly helpful when rearranging terms in an expression to make combining them easier.### Understanding the Commutative Property- You can think of it as saying \(a + b = b + a\). It doesn’t matter how you order them, the sum remains the same.- Similarly, for multiplication, \(a \times b = b \times a\).In the context of our expression \(2a + 4 + 3a + 5\), using the commutative property allows us to rearrange the terms as \(2a + 3a + 4 + 5\). By grouping like terms together, it becomes clearer which terms can be combined.Utilizing the commutative property can simplify a problem significantly, especially when dealing with complex expressions. It empowers you to rearrange terms to create order and support the identification and combining of like terms.