Problem 46 - Algebra A Combined Function

Question

Rewrite each sentence using mathematical symbols. Do not solve the equations. Twice the sum of 8 and a number is the difference of the number and 20 .

Step-by-Step Solution

Answer

The equation is \( 2(8 + x) = x - 20 \).

Step 1: Identify the Variables

First, we need to define the unknown quantity mentioned in the sentence. The phrase "a number" refers to an unknown value, which we can represent with a variable, usually denoted as \( x \).

Step 2: Translate the Sum into Symbols

The sentence mentions "the sum of 8 and a number." Mathematically, this is expressed as \( 8 + x \) where \( x \) is the variable we defined earlier.

Step 3: Express Twice the Sum

Since the sentence specifies "twice the sum of 8 and a number," we multiply the sum \( 8 + x \) by 2, resulting in \( 2(8 + x) \).

Step 4: Translate the Difference into Symbols

The sentence also mentions "the difference of the number and 20," which indicates subtraction. Using our variable, this can be written as \( x - 20 \).

Step 5: Write the Full Equation

According to the sentence, twice the sum equals the difference. Therefore, we set the expressions equal to each other to form the equation: \[ 2(8 + x) = x - 20 \]

Key Concepts

Variables

In algebra, a variable is a symbol used to represent an unknown value in mathematical expressions or equations. It's a fundamental concept, as it lets us work with numbers even when they aren't known yet. When we encounter a phrase like "a number" in a problem, we often use a variable, usually denoted as \( x \) or any letter, to stand in for the value we're trying to find. This makes it easier to perform algebraic operations and solve equations later on.
Variables can change, unlike constants, which always have a fixed value. For instance, in the phrase "a number plus eight," "a number" can be any value, and therefore, it is represented by a variable like \( x \). Understanding variables is key to solving algebraic problems, as they can appear in various parts of mathematical equations.

Mathematical Symbols

Mathematical symbols are the notations used to represent numbers, operations, relations, and other concepts in mathematics. They are the building blocks of math language and allow us to express complex concepts succinctly and efficiently.
Some common symbols include:

• \(+,-, \times, \div\): Represent operations like addition, subtraction, multiplication, and division.
• \(=\): Shows equality, indicating that two expressions represent the same value.
• \((, )\): Used to group expressions and define the order of operations.
• \(x\): Often used as a variable in algebraic expressions.

Using these symbols, we can translate words into mathematical language, enabling us to solve problems systematically and accurately.

Equations

An equation is a mathematical statement that asserts the equality of two expressions. It is composed of two parts, both containing numbers and variables, linked by the equals sign \( = \).
In our example, we derived the equation \[ 2(8 + x) = x - 20 \]. This particular equation tells us that twice the sum of 8 and the unknown number equals the difference between the same unknown number and 20. By creating an equation, we have a foundation for solving the problem, either by isolating the variable or using other algebraic methods.
Equations can be simple, like \( x + 2 = 5 \), or more complex, involving multiple terms and variables. They are vital in problem-solving as they provide a precise way to describe and solve mathematical problems.

Translation into Symbols

Translating sentences into mathematical symbols is a crucial step in solving word problems in mathematics. This involves identifying key phrases and translating them into mathematical expressions using appropriate symbols and operations.
For our given sentence: "Twice the sum of 8 and a number is the difference of the number and 20," we follow these steps:

• Identify phrases like "twice the sum," which translates to multiplying a sum by 2.
• "The sum of 8 and a number" becomes \( 8 + x \).
• "Twice the sum" translates to \( 2(8 + x) \).
• Identify "the difference of the number and 20," which is expressed as \( x - 20 \).

This translation allows us to formulate the equation \( 2(8 + x) = x - 20 \). Translating words into math symbols is a step-by-step process that turns verbal problems into equations we can work with, making them more manageable to solve.