Problem 18 - 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. $$-8 a+3 a+12+1$$

Step-by-Step Solution

Answer

The expression simplifies to \(-5a + 13\).

Step 1: Identify Like Terms

In the expression \(-8a + 3a + 12 + 1\),identify terms that are alike. Terms \(-8a\) and \(3a\) are like terms because they both contain the variable \(a\). The numbers \(12\) and \(1\) are also like terms because they are constant terms.

Step 2: Group Like Terms

Use the commutative property to group like terms together:\((-8a + 3a) + (12 + 1)\). This allows us to simplify each group individually.

Step 3: Simplify Like Terms

Simplify each group of like terms. For the terms with \(a\):\(-8a + 3a = -5a\).For the constant terms:\(12 + 1 = 13\).Thus, the expression simplifies to:\(-5a + 13\).

Key Concepts

Understanding Like Terms

In algebra, the concept of 'like terms' is crucial for simplifying expressions. Like terms are terms that have exactly the same variables raised to the same power. This means you can combine them easily because they essentially represent the same thing.
For example, in the expression

• \(-8a\) and \(3a\) are like terms because both terms contain the variable \(a\).
• On the other hand, \(12\) and \(1\) are like terms because they are both constants, meaning they do not have any variables attached to them.

When simplifying expressions, always start by identifying like terms. Group them so that you can add or subtract the coefficients straightforwardly. This is an essential step for making expressions easier to work with. Remember that mixing terms with different variables or powers is not possible while combining like terms.

Using the Commutative Property

The commutative property is a fundamental rule in algebra that makes it easier to rearrange terms in an expression without changing its value. This property applies to both addition and multiplication, and it states that the order in which you add or multiply numbers does not matter.
For example, if you have an expression like

• \(-8a + 3a + 12 + 1\),
• thanks to the commutative property, you can rearrange it to i. \((-8a + 3a) + (12 + 1)\),

making it easier to group and simplify like terms. The commutative property is particularly helpful when expressions get more complicated, allowing you to organize terms in a way that simplifies the arithmetic. This property is a powerful tool for keeping expressions neat as you solve more complex problems.

Algebraic Expressions Simplified

Algebraic expressions can seem a bit daunting at first, but with a little practice, they become more understandable. An algebraic expression is simply a mathematical phrase that can contain numbers, variables, and operations.
A key part of working with these expressions is knowing how to simplify them. Simplifying involves combining like terms and using properties such as the commutative property to rearrange and reduce the expression. Following these steps helps make sure you're addressing the expression in a logical and systematic way.

• For instance, with the expression \(-8a + 3a + 12 + 1\),
• you can easily simplify it to \(-5a + 13\),

by first grouping and then combining the like terms. Simplifying algebraic expressions allows for a clearer understanding and makes solving equations or evaluating expressions much more manageable.