Problem 362 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ -4(q-7) $$

Step-by-Step Solution

Answer

-4q + 28

Step 1: Identify the expression inside the parentheses

The given expression is \( -4(q-7) \). Here, \ q-7 \ is the expression inside the parentheses.

Step 2: Apply the Distributive Property

According to the Distributive Property, multiply each term inside the parentheses by the term outside. This means multiplying \ -4 \ by each term inside \ q-7 \: \[ -4 \times q \] and \ -4 \times -7 \.

Step 3: Perform the multiplication for each term

First, multiply \ -4 \ by \ q \: \( -4q \). Next, multiply \-4 \ by \ -7 \ which gives \ 28 \: \( -4 \times -7 = 28 \).

Step 4: Combine the results

Combine the results of the multiplications: \( -4q \) and \( 28 \). Thus, \ -4(q-7)= -4q + 28 \.

Key Concepts

algebraic expressions

Algebraic expressions are a crucial part of algebra and mathematics in general. These expressions consist of variables, constants, and arithmetic operations that combine them.
An example of an algebraic expression is: \(-4(q-7)\).
There are a few key components to understand:

• Variables: Letters like \(q\) which can represent any number.
• Constants: Numbers like -4 and 7, which always have the same value.
• Operations: Symbols like \( +, -, \times, / \) used to combine variables and constants.
To make algebraic expressions useful, we perform operations that simplify them. This brings us to the next concept.

simplifying expressions

Simplifying expressions means making them easier to understand or work with. Using rules from algebra, we can turn complex expressions into simpler ones.
Consider the example \(-4(q-7)\). We will use the Distributive Property to simplify this. Simplification involves several steps, such as:

• Identify: Finding the parts of the expression within parentheses or other groupings.
• Distribute: Applying the distributive property, where a term outside the parentheses multiplies every term inside.
• Combine: Putting together results from the distributive step to form a simpler expression.
Once simplified, the expression is easier to work with or further manipulate, as seen with \(-4(q-7)= -4q + 28\). Now, let's focus on the actual multiplication involved.

multiplication of terms

Multiplication of terms is an essential operation when simplifying algebraic expressions. It involves multiplying coefficients (numbers) with variables or other numbers.
In our example, \(-4(q-7)\), we carry out two multiplications:

• \(-4 \times q\) yields \(-4q\).
• \(-4 \times -7\) yields \(28\) because multiplying two negatives gives a positive.
Therefore, applying these multiplication rules transforms \( -4(q-7) \) into \(-4q + 28\). Remember these tips when multiplying terms:

• Always pay attention to signs (positive or negative).
• Multiply constants with constants and coefficients with variables properly.
Mastering these steps will make working with algebraic expressions much simpler!