Problem 19 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The sum of three consecutive integers is \(279 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 92, 93, and 94.

Step 1: Define the Variables

Let's call the three consecutive integers \(x\), \(x+1\), and \(x+2\). Here, \(x\) is the first integer. By defining the sequence in this way, we ensure that the next two integers are consecutive.

Step 2: Set Up the Equation

The problem states that the sum of these three integers is \(279\). Therefore, we can set up the following equation: \(x + (x+1) + (x+2) = 279\).

Step 3: Simplify the Equation

Combine like terms in the equation: \(x + x + 1 + x + 2 = 279\), which simplifies to \(3x + 3 = 279\).

Step 4: Solve for x

Subtract 3 from both sides of the equation to isolate the term with \(x\): \[3x + 3 - 3 = 279 - 3\]This simplifies to: \[3x = 276\]Now, divide both sides by 3 to solve for \(x\):\[x = 276 / 3\]\[x = 92\]

Step 5: Find the Consecutive Integers

Now that we know \(x = 92\), we can find the three consecutive integers. They are \(92\), \(93\) (which is \(x+1\)), and \(94\) (which is \(x+2\)).

Key Concepts

Algebraic Equations

An algebraic equation is a mathematical statement indicating that two expressions are equal. In algebra, it is commonly represented by using the equality sign "=". When solving problems like finding consecutive integers, equations help in setting up relationships and uncovering unknown values. To solve such problems, one must first understand the structure of the equation in question:

• It consists of terms separated by operation signs such as addition, subtraction, multiplication, or division.
• Terms can include variables, constants, or coefficients.

For instance, in the exercise, the equation is formed by adding the expressions for the integers. Thus, the algebraic equation was: \[ x + (x+1) + (x+2) = 279 \] This setup allows us to work systematically towards solving for the unknown variable, "\(x\)", giving us a clear path to reveal the three integers that sum up to 279.

Integer Solutions

Finding the solution to an algebraic equation often involves determining integer values, especially when the problem relates to real-life scenarios like consecutive numbers. An integer is any whole number, positive, negative, or zero. In solving for consecutive integers:

• We expect the solution to yield whole numbers.
• Consecutive integers follow a single unit increment, as with 92, 93, and 94 in our exercise.

The process of finding integer solutions involves combining like terms, simplifying the equation, and isolating the variable to solve it. Once simplified, our equation was:\[ 3x + 3 = 279 \] From here, basic arithmetic operations lead us to the integer solution, \(x = 92\). Consequently, the consecutive integers are 92, 93, and 94. Ensuring the solution is sensible and consistent with the problem assures correctness.

Variable Definition

Variables play a crucial role in creating and solving algebraic equations. A variable represents an unknown or unspecified number and helps us set up equations that solve specific problems. Think of variables as placeholders that allow flexibility in mathematical expression and solutions. In the given exercise, defining the variable "\(x\)" correctly is the key first step:

• Start with labeling the initial integer as \(x\).
• The immediate subsequent values become \(x+1\) and \(x+2\), ensuring all values are expressed in terms of \(x\).

By setting the first integer as \(x\), the sequence maintained integrity as consecutive values were generated. This method simplifies both the setup and solving phases, transforming a word problem into a clear algebraic form. Proper variable definition streamlines obtaining the unknowns and confirming the accuracy of the entire solution.