Problem 52 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Find three consecutive odd integers whose sum is 291 .

Step-by-Step Solution

Answer

The three consecutive odd integers are 95, 97, and 99.

Step 1 - Define Variables

Let the first odd integer be represented by the variable x. Since the numbers are consecutive odd integers, the second and third integers can be represented as x+2 and x+4 respectively.

Step 2 - Set Up the Equation

The sum of the three consecutive odd integers is 291. Therefore, the equation can be set up as: x + (x+2) + (x+4) = 291

Step 3 - Combine Like Terms

Combine the like terms on the left side of the equation: x + x + 2 + x + 4 = 291 This simplifies to: 3x + 6 = 291

Step 4 - Solve for x

Isolate the variable x by subtracting 6 from both sides of the equation: 3x + 6 - 6 = 291 - 6 This simplifies to: 3x = 285 Next, divide both sides by 3: 3x/3 = 285/3 Therefore, x = 95

Step 5 - Find the Consecutive Integers

The first odd integer is x = 95. Using x, find the next two odd integers: The second integer is x + 2, which is: 95 + 2 = 97 The third integer is x + 4, which is: 95 + 4 = 99

Step 6 - Verify the Solution

Add the three integers to ensure they sum to 291: 95 + 97 + 99 = 291 Therefore, 95, 97, and 99 are the correct integers.

Key Concepts

Number Word Problems

Number word problems are a common type of math problem that requires you to find unknown numbers based on given conditions. In this case, the exercise asks to find three consecutive odd integers whose sum equals 291. Word problems typically provide a scenario or condition that necessitates an equation for solving. Turning the word problem into mathematical expressions is a critical step in solving them.

Here, the given condition is about consecutive odd integers, which are integers differing by 2 units each. Begin by defining a variable to represent the first unknown number, for instance, let the first odd integer be represented by 'x'. Then, use this variable to express the remaining integers in the sequence.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. In this example, after setting up the equation from the word problem as: \( x + (x+2) + (x+4) = 291 \),

First combine like terms: \( x + x + x + 2 + 4 = 291 \), which simplifies to \( 3x + 6 = 291 \). This equation needs to be solved for 'x'.

To isolate 'x', first eliminate the constant term by subtracting 6 from both sides: \( 3x + 6 - 6 = 291 - 6 \), yielding \( 3x = 285 \). Next, solve for 'x' by dividing both sides by 3: \( x = 95 \).

Now, 'x' represents the first integer. With this value, you can find the remaining integers by adding 2 and 4 respectively to 'x'.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving variables and constants combined using operators like addition, subtraction, etc. The first step in many math problems is forming an algebraic expression from the problem's text.

In forming expressions for this problem, let the first integer be 'x'. The next two consecutive odd integers will be \( x + 2 \ \text{ and } x + 4 \).
These expressions help to set up the equation that sums up the integers: \( x + (x+2) + (x+4) \).

Working with algebraic expressions is a foundational skill in algebra, and comprehending their structure helps you solve equations efficiently.

Basic Algebra

Basic Algebra includes understanding and manipulating mathematical symbols and expressions to find unknowns. Here's a brief approach:

- Define variables for unknown quantities, as shown by letting \( x \) be the first odd integer.
- Transition the word problem into an equation. Here, it’s \( x + (x+2) + (x+4) = 291 \).
- Simplify and combine like terms: \( 3x + 6 = 291 \).
- Isolate the variable: First subtract 6 from both sides and then divide by 3 to solve for 'x'.
- Use the value found to determine any further required values (like the second and third integers here as \( 97 \) and \( 99 \)).

Mastering these basics is crucial for tackling more complicated algebra problems in the future.