Problem 43 - Elementary Algebra

Question

Simplify. $$ (-5) 2+32-42+2 \cdot 7 $$

Step-by-Step Solution

Answer

The simplified expression is -6.

Step 1: Apply the Distributive Property

Let's break down the expression: \[ (-5) \cdot 2 + 32 - 42 + 2 \cdot 7 \]The first operation to perform is the multiplication: \[ (-5) \cdot 2 = -10 \]And the multiplication: \[ 2 \cdot 7 = 14 \]Now replace them to simplify the expression: \[ -10 + 32 - 42 + 14 \]

Step 2: Perform the Addition and Subtraction

Now we perform addition and subtraction from left to right. Start with the first two terms:\[ -10 + 32 = 22 \]Then use the result to subtract the next term:\[ 22 - 42 = -20 \]Finally, add the last term:\[ -20 + 14 = -6 \]

Step 3: Write the Final Answer

After simplifying and performing all the operations, the simplified expression is:\[ -6 \]

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle that helps in simplifying expressions. It is particularly useful when you need to multiply a number across a sum inside parentheses. Consider the expression \((-5) \cdot 2 + 32 - 42 + 2 \cdot 7\). Here, the distributive property allows you to break down the multiplication: \((-5) \cdot 2\) and \(2 \cdot 7\). This means you can treat each multiplication separately and replace the original numbers with their products.
For example:

• \((-5) \cdot 2\) gets simplified to \(-10\).
• \(2 \cdot 7\) gets simplified to \(14\).

After applying multiplication through the distributive property, the expression becomes: \(-10 + 32 - 42 + 14\). This step makes the rest of the calculations easier to handle.

Order of Operations

Understanding the order of operations is crucial when simplifying expressions. The conventional order of operations is often remembered by the acronym PEMDAS — Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In our problem \((-10 + 32 - 42 + 14\)), multiplication was completed in the previous step using distributive property.
This leaves us with addition and subtraction, which we perform from left to right:

• First, we add \(-10\) and \(32\) to get \(22\).
• Next, we subtract \(42\) from \(22\) resulting in \(-20\).
• Finally, we add \(14\) to \(-20\) to get \(-6\).

Following this sequence ensures a correct and simplified solution.

Addition and Subtraction

Addition and subtraction are the last operations when simplifying expressions, following multiplication and division as per the order of operations. Once you reach this step, focus on handling each operation sequentially from left to right.
In the expression \(-10 + 32 - 42 + 14\), perform addition and subtraction step-by-step:

• Add \(-10\) and \(32\) to arrive at \(22\).
• Subtract \(42\) from \(22\), resulting in \(-20\).
• Finally, add \(14\) to \(-20\), which simplifies to \(-6\).

This straightforward approach is efficient and prevents any mistakes that can arise from skipping steps or errors in sequencing the operations. The goal is to achieve a single final answer without any remaining addition or subtraction to perform.