Problem 18 - Prealgebra

Question

Define a variable. Then write an equation and solve. The sum of 7 and a number is \(23 .\)

Step-by-Step Solution

Answer

The unknown number is 16.

Step 1: Define the Variable

To start, define the unknown number as a variable. Let's call this number \( x \). This gives us a way to work with the unknown quantity.

Step 2: Set Up the Equation

Based on the problem statement, "The sum of 7 and a number is 23," we can translate this into an algebraic equation. This becomes: \( 7 + x = 23 \).

Step 3: Solve for x

To find the value of \( x \), we need to isolate it on one side of the equation. Subtract 7 from both sides of the equation to get: \( x = 23 - 7 \).

Step 4: Simplify the Equation

Perform the subtraction on the right side of the equation to solve for \( x \): \( x = 16 \).

Key Concepts

Variable Definition

When tackling algebra problems, the first step is often defining variables. A variable is simply a symbol, often a letter, that represents a number we do not know yet. Think of it as a placeholder for this unknown number. In our exercise, we defined the variable as \( x \).
This approach helps us translate words into mathematical language, making it easier to manipulate and solve problems.
Here's why variables are important:

• They simplify complex problems by reducing them to basic arithmetic operations.
• Variables allow you to generalize problems into equations, making it possible to find solutions by applying standard methods.
• They make patterns and relationships within the problem clearer and more organized.

Equation Solving

Once a variable has been defined, the next step is to set up an equation. An equation is a statement that expresses the equality of two mathematical expressions. In our exercise, the equation is \( 7 + x = 23 \).
This equation states that the sum of 7 and an unknown number (\( x \)) is equal to 23.
Solving an equation involves finding the value of the variable that makes the equation true.

• First, perform operations to isolate the variable on one side of the equation.
• In this case, subtract 7 from both sides to get \( x = 23 - 7 \).
• Simplify to find the solution: \( x = 16 \).

To ensure your solution is correct, you can substitute \( 16 \) back into the original equation to see if both sides equal 23.

Unknown Number

Understanding the concept of an unknown number is crucial in algebra. An unknown number is simply a number you don't know but want to find. This is what makes problems like our exercise both interesting and solvable.
In our case, the unknown number was represented by \( x \), and the task was to identify its value using the equation.
Here's how to think about unknown numbers:

• They are often represented by variables, which act as versatile placeholders.
• Solving for an unknown involves using mathematical operations to transform the equation until the unknown number is isolated.
• Our goal is always to determine its precise value, which in this exercise, turned out to be 16.

Unknown numbers make algebra dynamic and engaging, offering a way to solve problems that depend on conditions or situations we can't predict.