Problem 83 - Algebra A Combined Function

Question

Translate each phrase; then simplify. See Example 22. Find the sum of \(-31,-9,\) and 30 .

Step-by-Step Solution

Answer

The sum of -31, -9, and 30 is -10.

Step 1: Translate the Phrases into Numbers or Expressions

First, identify the numbers and the operation described by the phrase. In this case, we have the numbers \(-31, -9,\) and \(30\), and we are asked to find their sum. This translates to the expression:\[-31 + (-9) + 30\]

Step 2: Add the Numbers Together

Calculate the sum by adding the numbers in the expression. Start by adding \(-31\) and \(-9\):- \(-31 + (-9) = -40\)Next, add \(30\) to the result:- \(-40 + 30 = -10\).

Key Concepts

Algebraic Expressions

In math, an algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division). It represents a particular value. For instance, in the problem of finding the sum of \(-31, -9,\) and \(30\), we create an expression. Here, the expression is \(-31 + (-9) + 30\). This means you are stacking numbers according to their respective algebraic signs and operations. These expressions act like a sentence in math, allowing you to express and communicate mathematical ideas.
Expressions are essential in illustrating relationships in math problems. When dealing with integer addition, they help translate word problems into numbers and symbols that can be calculated. This requires an understanding of not just the numbers, but also the operations connecting them. Therefore, having clarity on what an expression represents is key to solving math problems efficiently.

Negative Numbers

Negative numbers are numbers less than zero, denoted with a minus sign (\(-\)). They appear in various real-world contexts, like temperatures below freezing or debts. In an expression, negative numbers follow similar arithmetic rules to positive numbers. When adding negative numbers, it's important to remember these rules:

• Additive Opposites: Any number with its opposite results in zero. For example, \(-31 + 31 = 0\).
• Combined Negatives: When two negative numbers are added together, their absolute values are added, and the result is negative. Thus, \(-31 + (-9) = -40\).
• Positives and Negatives: When adding a positive number to a negative number, subtract the smaller absolute value from the larger and take the sign of the larger. For example, \(-40 + 30 = -10\).

Understanding how these concepts work helps streamline calculations, especially in expressions combining positive and negative integers. Visualizing them on a number line can also assist in comprehending their relationships.

Problem-solving Steps

Solving math problems often requires a systematic approach or a series of steps. This structured method helps in understanding and solving the problem effectively and efficiently.
To solve the problem of finding the sum of \(-31, -9,\) and \(30\), follow these steps:

• Identify the numbers and operation: Recognizing what the task requires is crucial. Here, you identify the numbers \(-31, -9,\) and \(30\), and determine you need to find their sum.


• Translate into a mathematical expression: Take the identified numbers and operation, then write them in an equation, like \(-31 + (-9) + 30\).


• Calculate step-by-step: Begin by solving smaller parts of the expression. Add \(-31\) and \(-9\) first, which equals \(-40\). Proceed by adding \(30\) to \(-40\), resulting in \(-10\).

This systematic approach of breaking down what the problem is asking for, converting it into a manageable form, and solving piece by piece, can be applied to various types of math problems. The key is to be clear and organized in each step of the problem-solving process.