Problem 49 - 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. $$-2(8-10)+3(4-9)$$

Step-by-Step Solution

Answer

The simplified expression is \(-11\).

Step 1: Parentheses

Identify and simplify the expressions within the parentheses first. For \(8 - 10\), the result is \(-2\). For \(4 - 9\), the result is \(-5\). So the expression becomes: \[-2(-2) + 3(-5)\]

Step 2: Multiplication

Next, perform the multiplication operations. Multiply \(-2\) by \(-2\) to get \(4\), and \(3\) by \(-5\) to get \(-15\). So the expression becomes: \[4 - 15\]

Step 3: Addition/Subtraction

Perform the addition or subtraction from left to right. Combine \(4\) and \(-15\) to get \(-11\).

Key Concepts

Parentheses

The first step in simplifying mathematical expressions is to tackle any calculations inside parentheses. This is crucial due to the order of operations, commonly remembered by the acronym PEMDAS (or BODMAS in some regions), which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. Calculating expressions within parentheses ensures that all operations enclosed are completed before handling the rest of the expression.
In our given expression \(-2(8-10)+3(4-9)\), we start by simplifying within the parentheses:

• For \(8 - 10\), subtracting 10 from 8 gives us \(-2\).
• For \(4 - 9\), subtracting 9 from 4 produces \(-5\).

Once these internal calculations are complete, each parentheses can be replaced with a single value. The expression now looks simpler and is ready for the next operations: \[-2(-2) + 3(-5)\].
By handling parentheses first, you ensure that your calculations remain correct and methodical, paving the way for the correct solution through subsequent steps.

Addition and Subtraction

Addition and subtraction are operations that come last in the order of operations. While this might seem straightforward, it's crucial to carry out these operations from left to right. After simplifying an expression, perform addition and subtraction only after dealing with parentheses and any multiplication or division.
For our expression, we ended with this after the multiplication step: \[4 - 15\].
Perform the subtraction to reach the final answer:

• Start with 4 and subtract 15.
• This gives a result of \(-11\), which is the final simplified value of the expression.

The rule to remember here is that consistent left to right calculation ensures accuracy and prevents error when multiple addition and subtraction operations are involved in a larger expression.
By maintaining this sequence, you can confidently navigate through complex problems to reach a correct answer.

Multiplication and Division

Multiplication and division operations come after solving within parentheses, according to the order of operations. These two operations should be carried out from left to right as they appear in the expression. This step ensures you're simplifying the expression efficiently and correctly.
In the example expression, after replacing the parentheses by their respective values, \(-2(-2) + 3(-5)\), we perform the multiplications:

• First, multiply \(-2\) by \(-2\). Multiplying two negative numbers gives a positive result, resulting in \(4\).
• Next, multiply \(3\) by \(-5\), resulting in \(-15\).

These calculations change our expression to \[4 - 15\].
It’s important to treat multiplication and division as equal priority operations, always working through them from left to right across the expression. This methodical approach prevents errors and streamlines the process toward the final solution.