Problem 30 - 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. $$8 y+1+6 y$$

Step-by-Step Solution

Answer

The simplified expression is \(14y + 1\).

Step 1: Identify Like Terms

In the expression \(8y + 1 + 6y\), identify the like terms. The terms \(8y\) and \(6y\) are similar because they both contain the variable \(y\). The number 1 is a constant term, and it does not have any like terms to combine with.

Step 2: Use the Commutative Property to Rearrange

Rearrange the terms using the commutative property, which allows us to change the order of addition. The expression can be rewritten as \(8y + 6y + 1\) to make it easier to combine like terms.

Step 3: Combine Like Terms

Add the coefficients of the like terms \(8y\) and \(6y\). This gives you \((8 + 6)y = 14y\). The expression now becomes \(14y + 1\).

Step 4: Simplified Expression

Since there are no other like terms to combine, the simplified expression is \(14y + 1\).

Key Concepts

Commutative Property

The commutative property is an essential algebraic principle that allows us to rearrange numbers or algebraic expressions to make calculations simpler. In basic terms, it states that the order in which you add or multiply numbers does not change the result.
For example, if you have an expression like \(a + b\), you can swap the numbers around to write it as \(b + a\). Both expressions will yield the same result when calculated. This property becomes incredibly useful when simplifying algebraic expressions because it allows us to rearrange terms, making it easier to identify and combine like terms.

• Helps rearrange terms in expressions.
• Makes combining like terms simpler.
• Useful in both addition and multiplication scenarios.

In our exercise, we used the commutative property to shift the terms \(8y + 1 + 6y\) into the order \(8y + 6y + 1\), which simplifies the process of combining like terms.

Like Terms

"Like terms" are terms in an expression that have identical variables and can thus be combined. Understanding this concept is crucial in simplifying expressions.
For instance, in the expression \(8y + 1 + 6y\), the terms \(8y\) and \(6y\) are like terms because they both include the variable \(y\). However, the term \(1\) is not similar to \(8y\) or \(6y\) because it does not contain a \(y\).

• Like terms must have the same variable(s).
• Only like terms can be combined.
• Checking for like terms simplifies expression reduction.

Recognizing and pairing these terms allows for the expression to be reduced more quickly to its simplest form.

Combining Like Terms

Once you identify like terms, the next step is combining them into a single term to simplify the expression further. This is achieved by adding or subtracting their coefficients while keeping the variable part unchanged.
In our example, after rearranging the terms as \(8y + 6y + 1\), we combine the coefficients of the like terms \(8y\) and \(6y\):

• Add their coefficients: \(8 + 6 = 14\).
• Write the new term with the common variable: \(14y\).

This reduces the given expression to \(14y + 1\), which is the simplified form. Combining like terms is a fundamental process that helps in managing and simplifying complex expressions efficiently.