Problem 88 - Algebra A Combined Function

Question

Simplify each expression. See Section \(1.8 .\) \(8(z-6)+7 z-1\)

Step-by-Step Solution

Answer

The simplified expression is \(15z - 49\).

Step 1: Distribute the Coefficient

First, apply the distributive property to the expression: \[ 8(z-6) \]This means multiplying 8 with each term inside the parentheses. \[ 8(z) - 8(6) = 8z - 48 \]So, the expression becomes: \[ 8z - 48 + 7z - 1 \]

Step 2: Combine Like Terms

Now, identify and combine the like terms from the expression. We have two like terms that involve \(z\):\[ 8z + 7z \]Combine them:\[ 15z \]Similarly, combine the constant terms:\[ -48 - 1 \]Simplifying this gives:\[ -49 \]Now the expression is:\[ 15z - 49 \]

Key Concepts

Distributive Property

The distributive property is a useful algebraic rule that helps us simplify expressions by allowing us to multiply a single term by each term within a set of parentheses. Consider the expression \(8(z - 6)\). Here, we're using the distributive property.

This signifies that we will multiply 8 with both \(z\) and \(-6\). Let's break it down further:

• First, multiply \(8\) by \(z\), resulting in \(8z\).
• Then, multiply \(8\) by \(-6\), giving us \(-48\).

So, distributing \(8\) through the parentheses transforms it to \(8z - 48\).

Distributive property is quite handy, especially when dealing with expressions that involve parentheses, because it allows you to simplify the terms and eventually makes solving equations easier.

Combining Like Terms

After using the distributive property, the expression might still have several terms that can be made even simpler. This involves combining like terms. Like terms are terms that have the same variable part. In the expression \(8z - 48 + 7z - 1\), let's identify them:

• The terms \(8z\) and \(7z\) are like terms because they both have \(z\).
• The constants \(-48\) and \(-1\) are also like terms because they don't have any variables associated with them.

Combine \(8z\) and \(7z\) to get \(15z\). For the constant terms \(-48\) and \(-1\), adding them gives \(-49\).

Combining like terms is essentially addition and subtraction of the coefficients of these terms, which makes the expression more refined and easier to work with.

Simplification

Simplification is the final stage in refining an algebraic expression to its simplest form. After distributing and combining like terms, as shown earlier, we're left with \(15z - 49\). Let's take a closer look at this process:

• Simplification means ensuring there are no more like terms left to be combined.
• It also checks that there's minimal redundancy in the expression.

In cases like \(15z - 49\), this is as simple as it can get because:

• There are no more like terms to combine.
• The expression doesn't have any factors that can be simplified further without changing its value.

Simplification not only makes the expression more elegant but also easier to interpret and use in further calculations or real-world applications.