Problem 47 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$3(5-8)+4(6-7)$$

Step-by-Step Solution

Answer

The simplified expression is \(-13\).

Step 1: Apply Parentheses Simplification

First, simplify the expressions inside the parentheses in the expression \(3(5-8)+4(6-7)\). Calculate \(5-8\) to get \(-3\) and \(6-7\) to get \(-1\). So the expression simplifies to \(3(-3) + 4(-1)\).

Step 2: Multiplication

Next, perform the multiplication operations. Multiply \(3\) by \(-3\) to get \(-9\) and \(4\) by \(-1\) to get \(-4\). So now the expression is \(-9 + (-4)\), which is equivalent to \(-9 - 4\).

Step 3: Addition/Subtraction

Add the results from the multiplication step to find the final value. \(-9 - 4\) equals \(-13\).

Key Concepts

Parentheses Simplification

Understanding how to simplify expressions within parentheses is crucial in mathematics. Parentheses tell us which parts of an expression we need to focus on first, according to the order of operations. In the expression \(3(5-8)+4(6-7)\), our first step is to tackle the calculations inside the parentheses.

Here’s the process:

• Look inside the first set of parentheses \((5-8)\). Subtraction inside the parentheses gives you \(-3\).
• Next, look inside the second set of parentheses \((6-7)\). Again, perform the subtraction to get \(-1\).

By simplifying the parentheses statements first, we turn the original complex expression into a much simpler form: \(3(-3) + 4(-1)\). Note how the entire expression becomes easier to handle once the parentheses are simplified.

Multiplication Rules

Once the expressions inside the parentheses are simplified, the next step in the order of operations is multiplication. This step involves multiplying the numbers outside of the parentheses by the result we obtained from simplifying the parentheses.

Here's how it's done:

• Multiply the coefficient \(3\) by \(-3\). This results in \(-9\).
• Then, multiply \(4\) by \(-1\). This gives you \(-4\).

Multiplication could result in positives or negatives. A positive times a negative always gives a negative result. This step carefully transforms the expression into \(-9 + (-4)\), or equivalently \(-9 - 4\).

Recognizing and applying multiplication rules correctly simplifies the pathway to the final result.

Addition and Subtraction Rules

After completing the multiplication, the next task is to address the addition and subtraction components of the expression. This process is essentially about combining numbers to reach the final answer but must be done following the right rules.

For expressions like \(-9 - 4\), understanding how negatives interact is crucial. Here's what to consider:

• Adding two negative numbers is like adding their absolute values and then negating the result. So, for \(-9 - 4\), think of it like adding \(9\) and \(4\), then applying the negative sign, resulting in \(-13\).

Mastery of addition and subtraction, especially with negative values, is key to solving many algebraic expressions correctly. This final step wraps up the calculations and gives us the answer \(-13\).