Problem 53 - Elementary Algebra

Question

For the following problems, introduce a variable (any letter will do) and use appropriate algebraic symbols to write the given statement. Sixteen minus twice a number equals five.

Step-by-Step Solution

Answer

Question: Represent the statement "Sixteen minus twice a number equals five" as an algebraic equation using a variable. Answer: 16 - 2x = 5

Step 1: Introduce a variable

Let's use the variable x to represent the unknown number in the statement.

Step 2: Write the statement using algebraic symbols

We are given that "Sixteen minus twice a number equals five." This can be represented as an equation: 16 - 2x = 5.

Key Concepts

Variable Introduction

When solving algebraic problems, the first important step is to identify and introduce a variable.
A variable is simply a symbol, usually a letter, that stands for a number we do not yet know. In math, the most common letters used as variables are **x** and **y**, but honestly, any letter can serve as a variable.
In the given problem, we are tasked with understanding the phrase, "a number." Since we do not know what this number is, we introduce a variable to represent it. For this exercise, we use **x** as our variable.
Whenever you come across problems asking you to "introduce a variable," remember that you will be substituting in a letter to stand in for an unknown quantity. This variable will then be used in forming an equation to solve the problem.

Algebraic Equations

Once a variable has been introduced, the next step is to translate the problem statement into an algebraic equation.
Algebraic equations are mathematical sentences that use numbers, operation symbols, and variables to establish a relationship between quantities.
For example, the expression given is "Sixteen minus twice a number equals five." By using the variable **x** previously introduced to represent "a number," we can translate this statement into an algebraic equation.

• "Sixteen minus" becomes **16 -**
• "Twice a number" is represented as **2x**
• "Equals five" simplifies to **= 5**

So, the algebraic equation that represents the entire statement is: \[16 - 2x = 5\]Equations like this one allow us to find the unknown number by solving for **x**.

Problem-solving in Algebra

To solve an algebraic equation, we follow a series of steps to isolate the variable, which in this case is **x**.
Let's go through the process of solving the equation \(16 - 2x = 5\). To isolate **x**, we need to get it by itself on one side of the equation. Here's how:

• Subtract 16 from both sides of the equation to get rid of the constant on the left side:

\[16 - 16 - 2x = 5 - 16\]This simplifies to:\[-2x = -11\]

• Next, divide both sides by -2 to solve for **x**:

\[x = \frac{-11}{-2}\]The negative signs cancel out, leaving:\[x = \frac{11}{2}\]Remember, every step involves doing the same operation to both sides of the equation to maintain balance.
By performing these steps, you isolate the variable and solve the problem, finding the value of **x**.