Problem 29 - Elementary Algebra

Question

Set up an algebraic equation and then solve. The sum of three consecutive odd integers is 57 . Find the integers.

Step-by-Step Solution

Answer

The integers are 17, 19, and 21.

Step 1: Define the Variables

Let the first odd integer be \( x \). Since odd integers follow one another by a difference of 2, the next two consecutive odd integers can be defined as \( x + 2 \) and \( x + 4 \).

Step 2: Set Up the Equation

The problem states that the sum of these three integers is 57. Thus, the equation to represent this is:\[x + (x + 2) + (x + 4) = 57\]

Step 3: Simplify the Equation

Combine like terms in the equation:\[3x + 6 = 57\]

Step 4: Solve for x

Subtract 6 from both sides of the equation to isolate terms involving \( x \):\[3x = 51\]Next, divide each side by 3 to find \( x \):\[x = 17\]

Step 5: Find the Consecutive Integers

Now that we have \( x = 17 \), substitute \( x \) back to find the three integers:- First integer: \( 17 \)- Second integer: \( 17 + 2 = 19 \)- Third integer: \( 17 + 4 = 21 \)

Step 6: Verify the Solution

Check the sum of the integers to ensure it equals 57:\[17 + 19 + 21 = 57\]Since the equation holds true, our solution is verified.

Key Concepts

Algebraic Equation

In mathematics, an algebraic equation is a statement of equality that involves variables and constants. Equations are pivotal in conveying relationships involving numbers and unknowns. For the problem of consecutive odd integers, we formed an algebraic equation. These equations help us translate word problems into mathematical expressions.

Creating an algebraic equation involves identifying unknown values and defining them using variables. In our example, since we want to find three consecutive odd integers whose sum is 57, we start by letting the first integer be represented by the variable \( x \). This allows us to represent the subsequent integers as \( x+2 \) and \( x+4 \), reflecting the pattern of odd numbers we aim to sum up. This setup helps us construct a mathematical statement where all unknowns are connected logically.

Sum of Integers

The sum of integers is a fundamental concept that appears across various mathematical problems. In our scenario with consecutive odd integers, the sum is captured by adding all the values together to form an equation. This concept not only applies to odd integers but to any sequence of numbers.

To find the sum of consecutive odd integers, our goal was to add \( x \), \( x+2 \), and \( x+4 \) to get the sum of 57. Thus, we set up the expression \( x + (x+2) + (x+4) \) equal to 57. By treating this expression as a simple addition problem, we aim to reveal the value of \( x \), which represents the first odd integer. Understanding the sum involves evaluating how numbers relate to each other in sequences and the result they cumulatively produce.

Solving Equations

Solving equations is the process of determining the values of variables that satisfy the given mathematical statement. In our example, we rearrange the algebraic equation to isolate the variable of interest, allowing us to determine specific integer values.

In this consecutive odd integer problem, we start with the equation \( 3x + 6 = 57 \) derived from simplifying the sum of all the consecutive integers. To solve for \( x \), subtract 6 from both sides to get \( 3x = 51 \). Dividing each side by 3 gives the solution \( x = 17 \).

• Substituting back, \( x \) gives the first integer.
• By adding 2, we obtain the second integer.
• Adding 4 yields the third integer.

This step-by-step resolution ensures we reach the exact sequence of integers adding up to 57. Verifying by substituting the values into the original equation confirms the accuracy of the solution.