Problem 31 - 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. $$4 a-3-5 a+2 a$$

Step-by-Step Solution

Answer

The simplified expression is \(a - 3\).

Step 1: Identify Like Terms

In the expression, identify terms that have the same variables raised to the same power. The expression is \(4a - 3 - 5a + 2a\). The like terms here are \(4a\), \(-5a\), and \(2a\).

Step 2: Rearrange the Expression

Using the commutative property of addition, rearrange the terms so that like terms are next to each other. The expression rearranges to \(4a - 5a + 2a - 3\).

Step 3: Combine Like Terms

Add or subtract the coefficients of like terms. Combine \(4a\), \(-5a\), and \(2a\): \(4 - 5 + 2 = 1\), so the expression becomes \(1a - 3\).

Step 4: Simplify

Since multiplying by 1 doesn’t change a term, \(1a\) can be simplified to \(a\). Thus, the simplified expression is \(a - 3\).

Key Concepts

Understanding Like Terms

Identifying "like terms" is crucial when simplifying algebraic expressions. Like terms are terms in an expression that have the same variables raised to the same power. This means that the coefficients (the numerical parts) can be different, but the variable part must be identical. For example, in the expression \(4a - 5a + 2a\), the terms \(4a\), \(-5a\), and \(2a\) are like terms because they all contain the variable \(a\) raised to the first power.
To simplify, you focus on the coefficients. Add or subtract these coefficients, leaving the variable part unchanged. This process is much like combining apples with apples – since they share the same variable, they can be summed or subtracted. So, in the expression \(4a - 5a + 2a\), you would add \(4\), subtract \(5\), and add \(2\), ending up with \(1a\). If there were terms with different variables, like \(b\) or \(c\), they would not be combined with \(a\) terms.

Commutative Property in Algebra

The commutative property is a fundamental principle that facilitates simplifying algebraic expressions. This property states that the order in which two numbers or terms are added does not change the sum. In mathematical terms, \(a + b = b + a\). This property is especially useful when you need to organize an expression to easily combine like terms.
For instance, consider our original expression \(4a - 3 - 5a + 2a\). Using the commutative property, you can rearrange it to \(4a - 5a + 2a - 3\). This grouping of like terms – those with the variable \(a\) – allows you to focus on simplifying the expression by coefficient combination. The constant term, \(-3\), remains separate because it doesn't have a variable like \(a\).
Understanding and utilizing the commutative property ensures that expressions are neat and simplifies the process of combining like terms.

The Process of Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. This is achieved by combining like terms and removing any unnecessary complexity. First, identify and group like terms using the commutative property as needed. Once grouped, you can proceed to sum or subtract the coefficients of these like terms.
From our example, after rearranging, the expression was \(4a - 5a + 2a - 3\). By combining the coefficients \(4 - 5 + 2\), you simplify the expression to \(1a - 3\). The term \(1a\) further simplifies to \(a\) because multiplying by 1 does not change the value of \(a\).
In the final step, you are left with the simplified expression \(a - 3\). This process makes expressions easier to handle and read, paving the way for further algebraic manipulation or use in equations.