Problem 108 - Elementary Algebra

Question

The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 17 and 13.

Step 1: Define variables

Let the first number be denoted as \( x \). Since one number is 4 less than the other, let the second number be \( x - 4 \).

Step 2: Set up the equation

According to the problem, the sum of the two numbers is 30. Therefore, we can write the equation: \[ x + (x - 4) = 30 \]

Step 3: Simplify the equation

Combine like terms to simplify the equation: \[ 2x - 4 = 30 \]

Step 4: Solve for \( x \)

Add 4 to both sides of the equation to isolate the term with \( x \): \[ 2x - 4 + 4 = 30 + 4 \] which simplifies to \[ 2x = 34 \]. Next, divide both sides by 2: \[ x = 17 \].

Step 5: Find the second number

Since the second number is 4 less than the first number: \[ 17 - 4 = 13 \].

Key Concepts

solving equations

Solving equations involves finding the value of the variable that makes the equation true. In this problem, we start by writing an equation that represents the relationship described. Following this, we systematically apply mathematical operations to isolate the variable and find its value.
In our case, the initial equation is: \[ x + (x - 4) = 30 \]
To solve this, we perform a series of steps that will simplify and isolate the unknown variable. These steps include combining like terms, adding, subtracting, multiplying, or dividing both sides of the equation so that we gradually reveal the value of the variable.

variable definition

Defining variables is a crucial initial step in any word problem. It involves assigning letters to the quantities we need to find.
In the given problem, we are dealing with two numbers whose sum is 30, and one is 4 less than the other. We start by letting the first number be represented by the variable \( x \).
The problem tells us the second number is 4 less than the first, so we represent the second number as \( x - 4 \).
This helps in translating the word problem into a mathematical equation that we can solve.

combining like terms

Combining like terms is an important step in simplifying any equation. Like terms are terms that contain the same variable raised to the same power.
In the equation derived from our word problem: \[ x + (x - 4) = 30 \]
we have two \( x \) terms on the left side. To simplify, we combine these terms: \[ x + x - 4 = 30 \]
This results in: \[ 2x - 4 = 30 \].
By combining like terms, we reduce the complexity of the equation, making it easier to isolate the variable.

mathematical operations

Mathematical operations are the actions we perform on numbers and variables to solve equations. They include addition, subtraction, multiplication, and division.
In our example, we use these operations to isolate \( x \):
1. Add 4 to both sides to cancel out the -4: \[ 2x - 4 + 4 = 30 + 4 \]
This simplifies to: \[ 2x = 34 \]
2. Divide both sides by 2 to get: \[ x = 17 \]
Understanding these basic operations helps in systematically solving equations and finding the values of unknown variables.