Problem 18 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The sum of three consecutive integers is \(63 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 20, 21, and 22.

Step 1: Define Variables for Integers

Let's call the first integer \(x\). Since they are consecutive integers, the second integer will be \(x+1\) and the third integer will be \(x+2\).

Step 2: Set Up the Equation

The problem states that the sum of these three consecutive integers is \(63\). Therefore, we can set up the equation: \[ x + (x+1) + (x+2) = 63 \]

Step 3: Combine Like Terms

Combine the \(x\) terms and the constants on the left side of the equation: \[ 3x + 3 = 63 \]

Step 4: Solve for x

Subtract \(3\) from both sides of the equation to isolate terms with \(x\): \[ 3x = 60 \]Then, divide by \(3\) to solve for \(x\): \[ x = 20 \].

Step 5: Find the Three Consecutive Integers

Now that we have \(x = 20\), the three consecutive integers are:- First integer: \(x = 20\)- Second integer: \(x+1 = 21\)- Third integer: \(x+2 = 22\).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. For example, 7, 8, and 9 are consecutive integers because they come one right after the other in the number sequence. When working with consecutive integers in algebra, you will often be asked to find a series of numbers that meet certain conditions. These conditions are frequently expressed as equations. In our exercise, the task was to find three consecutive integers whose sum is 63. To express these integers in algebraic terms, we use a variable for the first integer, say, \( x \). The next integer will then be \( x+1 \), and the one after that \( x+2 \). Using variables like this is a common strategy in algebra to simplify and solve problems involving unknown numbers.

Solving Equations

Solving equations is a fundamental part of algebra. In our exercise, we set up an equation to find the values of three consecutive integers whose sum is 63. Let's walk through the process of solving such an equation.- **Write the Equation**: Start by writing down what the problem states in numbers and variables. In this example: \[ x + (x + 1) + (x + 2) = 63 \]- **Combine Like Terms**: Look for terms that can be combined to simplify the equation. Here, the \( x \) terms add up to \( 3x \) and the constants \( 1 \) and \( 2 \) add up to \( 3 \), giving us: \[ 3x + 3 = 63 \]- **Isolate the Variable**: You need to get \( x \) by itself to solve the equation. Subtract \( 3 \) from both sides: \[ 3x = 60 \]- **Solve for the Variable**: Finally, divide by \( 3 \) to find \( x \): \[ x = 20 \]By following these steps, you can systematically approach and solve algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition or multiplication). They are used to represent mathematical scenarios and are prevalent in algebra. In the problem given, an algebraic expression was set up to represent the sum of three consecutive integers:\( x + (x + 1) + (x + 2) \).Here is how the expression relates to the problem:

• Variables: \( x \), \( x+1 \), and \( x+2 \) are used to denote the unknown consecutive integers.
• Operations: The expression correctly sums them\( x + (x + 1) + (x + 2) \), aligning it with the given condition that their sum is 63.

These expressions serve as the building blocks to setting up equations which can then be solved to find unknown values. Understanding how to translate a word problem into an algebraic expression is critical in tackling math problems confidently.