Problem 28 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two numbers is 27 . One number is seven less than the other. Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 17 and 10.

Step 1: Identify Variables

Let the two numbers be represented by variables. Let one number be denoted as \( x \). Then, the other number can be represented as \( x - 7 \), since it is seven less than the first number.

Step 2: Set Up Equation

According to the problem, the sum of the two numbers is 27. The equation representing this relationship is: \[ x + (x - 7) = 27 \]

Step 3: Simplify Equation

Combine like terms in the equation: \[ x + x - 7 = 27 \] \[ 2x - 7 = 27 \]

Step 4: Solve for \( x \)

To isolate \( x \), first add 7 to both sides of the equation: \[ 2x - 7 + 7 = 27 + 7 \] \[ 2x = 34 \]Next, divide both sides by 2 to find \( x \): \[ x = \frac{34}{2} \] \[ x = 17 \]

Step 5: Find the Other Number

Using \( x = 17 \), calculate the other number: \[ x - 7 = 17 - 7 \] \[ x - 7 = 10 \]

Step 6: Solution Verification

Verify the solution by confirming the sum of the numbers is 27: \[ 17 + 10 = 27 \]The numbers are correct.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that involves finding the value of a variable that makes an equation true. In this problem, we start with an equation derived from a word problem. The sum of two unknown numbers is given, and one number is described in relation to the other.
To solve this equation:

• Identify the relationship between the numbers.
• Translate words into a mathematical equation.
• Combine like terms to simplify the equation.
• Use algebraic operations to isolate the variable and find its value.

This ensures that you systematically work towards the solution in a structured way.

Algebraic Expressions

Understanding algebraic expressions is crucial in transforming a word problem into an equation. An algebraic expression involves numbers, variables, and arithmetic operations.

In this problem:

• The first number is represented by the variable x.
• The other number is expressed as x - 7 (since it's seven less than the first number).

These expressions help us write an equation that we can solve to find the unknown numbers. Practice creating expressions from different word problems to strengthen your skills in algebra.

Variable Identification

Successful variable identification guides the entire solving process. A variable represents an unknown quantity in an equation. In this exercise, the following steps are used:

• Identify what needs to be found (`two unknown numbers`).
• Assign a variable, say x, to the first unknown number.
• Express the second unknown number in terms of the same variable (i.e., x - 7).

This process sets the foundation for writing the equation. Clearly identifying variables is essential for clarity and correctness in solving algebraic problems.

Equation Simplification

Simplifying equations is the process of reducing complexity by combining like terms and performing arithmetic operations. It transforms the equation into a simpler form, making it easier to solve. Here's how simplification works in this problem:

• Combine like terms: x + x - 7 = 27 becomes 2x - 7 = 27.
• Isolate the variable: Add 7 to both sides to get 2x = 34.
• Solve for the variable by dividing both sides by 2: x = 17.

Once simplified, it's straightforward to solve for the variable and find the numeric values. Practice simplifying various equations to master the process.