Problem 34 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 9(z+7)-15 $$

Step-by-Step Solution

Answer

The simplified expression is \(9z + 48\).

Step 1: Distribute 9 across the parentheses

Apply the distributive property to the term 9, multiplying it by each term inside the parentheses. This means you multiply 9 with both z and 7 independently: \[ 9(z) + 9(7) - 15 = 9z + 63 - 15 \]

Step 2: Combine Like Terms

Now, add or subtract any like terms to simplify the expression further. In this case, the like terms are the numbers 63 and -15. Perform the subtraction: \[ 9z + 63 - 15 = 9z + 48 \]

Step 3: Final Expression

After combining all like terms, the simplified expression is: \[ 9z + 48 \]

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that helps us simplify expressions involving parentheses. It states that you can multiply a number or variable outside the parentheses by each term inside the parentheses independently.
For instance, when faced with an expression like

• \(9(z+7)\),

the distributive property tells us to multiply 9 by each term inside the parentheses, which gives us:

• \(9 \cdot z + 9 \cdot 7 = 9z + 63\)

This step helps break down complex expressions into simpler terms, making it easier to work with and simplifying the solving process.

Combining Like Terms

Combining like terms is an essential technique in algebra used to simplify expressions further. Like terms are terms within an expression that share the same variables raised to the same power.

• For example, in the expression \(9z + 63 - 15\), \(63\) and \(-15\) are like terms because they are both constant numbers without variables.

To combine them, simply add or subtract these numbers:

• \(63 - 15 = 48\).

This process reduces the complexity of the expression, making it less cluttered and easier to understand or further manipulate. Remember, variables are not combined unless they are the same variable raised to the same power.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition or multiplication) that represent a value. These expressions can range from simple to complex, incorporation various mathematical operations and requiring simplification techniques to solve or express them in their most concise form.
For example, the expression:

• \(9z + 48\)

is derived from a more complex expression through processes like the distributive property and combining like terms.
Algebraic expressions can model real-world situations and problems, making them a versatile tool in mathematics. It’s crucial to get comfortable manipulating and simplifying these expressions to find solutions in equations.