Problem 51 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Find three consecutive odd integers whose sum is 171 .

Step-by-Step Solution

Answer

The three consecutive odd integers are 55, 57, and 59.

Step 1: Define the Variables

Let the first odd integer be represented by x. The next two consecutive odd integers can then be expressed as x + 2 and x + 4.

Step 2: Set Up the Equation

Form an equation based on the problem's description. The sum of the three consecutive odd integers is given as 171. Therefore, the equation is: \[ x + (x + 2) + (x + 4) = 171 \]

Step 3: Simplify the Equation

Combine like terms to simplify the equation: \[ x + x + 2 + x + 4 = 171 \] \[ 3x + 6 = 171 \]

Step 4: Solve for x

Isolate x by first subtracting 6 from both sides of the equation: \[ 3x = 165 \] Then, divide by 3: \[ x = 55 \]

Step 5: Find the Consecutive Odd Integers

Since x represents the first odd integer, the three consecutive odd integers are: \[ x = 55 \] \[ x + 2 = 57 \] \[ x + 4 = 59 \]

Step 6: Verify the Solution

Confirm that the sum of 55, 57, and 59 is indeed 171: \[ 55 + 57 + 59 = 171 \] The solution is correct.

Key Concepts

variables in algebra

In algebra, a variable is a symbol, usually a letter, that represents one or more numbers. For example, in the problem of finding three consecutive odd integers whose sum is 171, we use the variable 'x' to represent the first odd integer. This is helpful because we can express the next consecutive odd integers in terms of 'x', such as 'x + 2' and 'x + 4'. By defining these variables, we simplify the process of setting up and solving equations for unknown values.

solving equations

Solving equations is a critical skill in algebra that involves finding the value of variables that make an equation true. To solve the given problem, we first set up an equation using our defined variables:

• First odd integer = x
• Second odd integer = x + 2
• Third odd integer = x + 4

The problem states that their sum is 171, giving us the equation:

x + (x + 2) + (x + 4) = 171

We then simplify the equation by combining like terms:

x + x + 2 + x + 4 = 171 3x + 6 = 171

Next, isolate the variable 'x' by subtracting 6 from both sides:

3x = 165

Finally, divide by 3 to find the value of 'x':

x = 55

word problems in algebra

Word problems in algebra require translating a real-world situation into a mathematical equation. In the given exercise, we need to identify and write about consecutive odd integers. By defining our variable 'x' and expressing the next integers in terms of 'x', the problem becomes easier. Later, you can set up an equation and solve for 'x'. This method helps break down complex word problems into manageable steps. Remember to always:

• Define your variables clearly.
• Translate the words into mathematical expressions.
• Form an equation based on the problem description.
• Solve the equation step by step.

If you follow these steps, solving word problems becomes systematic and less intimidating.

integer operations

Understanding integer operations is essential in algebra. When working with consecutive numbers, addition and subtraction are prevalent. In this problem, we deal with consecutive odd integers, and we use addition to combine them:

• First odd integer: x
• Second odd integer: x + 2
• Third odd integer: x + 4

You then add these integers to form an equation:

x + (x + 2) + (x + 4)

When solving, you combine like terms to simplify the algebraic expression. The skills of addition, subtraction, and combining like terms are foundational for solving more complex algebraic equations. These integer operations make math problems manageable, enabling you to systematically solve and verify your solutions.