Problem 73 - Beginning and Intermediate Algebra

Question

Three times a number is equal to 8 more than twice the number.

Step-by-Step Solution

Answer

The number is 8.

Step 1: Understand the problem

Let's say the number we need to find is represented by the variable \( x \). The problem states that three times this number is equal to 8 more than twice the number.

Step 2: Set up the equation

Translate the verbal statement into an algebraic equation: Three times the number is equal to 8 more than twice the number. This can be written as: \[ 3x = 2x + 8 \]

Step 3: Isolate the variable

To find the value of \( x \), first subtract \( 2x \) from both sides of the equation: \[ 3x - 2x = 8 \]

Step 4: Simplify the equation

Simplify the left side of the equation: \[ x = 8 \]

Step 5: Solution

The value of the number is \( 8 \).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In the given exercise, we use the variable \( x \) to represent the unknown number we need to find. By defining and translating the problem statement, we create an algebraic expression that represents the scenario. For example, 'three times a number' translates to \( 3x \), and '8 more than twice the number' translates to \( 2x + 8 \). Using variables allows us to set up flexible expressions that facilitate solving equations.

Isolating the Variable

Isolating the variable means rearranging the equation to get the variable alone on one side. This step is crucial for solving for the unknown. In our exercise, we have the equation \( 3x = 2x + 8 \). To isolate \( x \), we need to perform operations that simplify the equation without changing its equality. By subtracting \( 2x \) from both sides of the equation, we get \( 3x - 2x = 8 \). Now, the variable \( x \) is isolated on one side, making it easier to determine its value.

Simplifying Equations

Simplifying equations involves performing mathematical operations to reduce them to their simplest form. This can involve combining like terms, reducing fractions, or simply performing basic arithmetic operations. In our step-by-step solution, after isolating the variable, we end up with the equation \( x = 8 \). Here, no additional simplification is required because \( x \) is already isolated and solved. The goal is always to achieve the simplest form, so the solution becomes clear and straightforward. Simplifying equations helps ensure that the solution is accurate and easy to understand.