Problem 53 - Algebra for College Students

Question

For Problems \(51-66\), use an algebraic approach to solve each problem. Find three consecutive integers whose sum is 42 .

Step-by-Step Solution

Answer

The integers are 13, 14, and 15.

Step 1: Define Variables

Let's define the first of the three consecutive integers as \( x \). Consequently, the next two integers can be described as \( x+1 \) and \( x+2 \).

Step 2: Set Up the Equation

The sum of these three consecutive integers is given as 42. Therefore, the equation is: \[ x + (x+1) + (x+2) = 42 \]

Step 3: Simplify the Equation

Combine like terms in the equation: \[ 3x + 3 = 42 \]

Step 4: Solve for x

Subtract 3 from both sides to isolate the term with \( x \): \[ 3x = 39 \] Now, divide by 3 to solve for \( x \): \[ x = 13 \]

Step 5: Find Consecutive Integers

With \( x = 13 \), the three consecutive integers are: \( 13 \), \( 14 \), and \( 15 \).

Key Concepts

Understanding Consecutive Integers

In mathematics, consecutive integers are numbers that follow each other in order without any gaps. Examples include numbers like 13, 14, and 15. They are sequenced one after the other without skipping a number. When dealing with problems involving consecutive integers, you can define them using an algebraic expression. If you let the first integer be represented as \( x \), then the next consecutive integers would simply be \( x+1 \), \( x+2 \), and so on.

This simple pattern allows you to express multiple numbers using just one variable, making it easier to set up and solve equations.

Approaching Equation Solving

Equation solving is a key skill in algebra. It's the process used to find unknown values that satisfy given equations. In our case, finding three consecutive integers whose sum is given involves solving an equation. By setting up an equation properly, you can determine the unknown value, which represents our consecutive integers.

Remember to follow these steps when solving equations:

• Write down all known information and create an equation based on that.
• Combine like terms if necessary and manipulate the equation to isolate the variable (in this case, \( x \)).
• Solve the equation step by step, checking your work as you go.

This structured approach helps ensure accuracy when finding solutions.

Defining Your Variable

Variable definition is an important first step in solving algebraic problems. A variable acts as a placeholder for unknown values. It is often represented by letters such as \( x \), \( y \), or \( z \). Before solving a given problem, you will define a variable to represent the unknown quantity. For consecutive integers, starting with \( x \) helps establish the subsequent integers as \( x+1 \) and \( x+2 \).

This method of defining variables forms the basis for writing your equation. It's the initial groundwork upon which further calculations are done. Make sure the definition aligns with the context of the problem for clarity and correctness.

Simplifying the Equation

Simplification in algebra helps make equations easier to solve by combining like terms and reducing complexity. Once you have an equation set up with defined variables, the next step is to simplify. For instance, in our equation \( x + (x+1) + (x+2) = 42 \), identify and add up all similar terms like the \( x \) terms, resulting in \( 3x \), and the constant terms, resulting in \( +3 \).

The simplified form, \( 3x + 3 = 42 \), is far easier to handle. Simplifying allows you to quickly see how modification and further manipulation can lead you to isolate and solve for \( x \). This step condenses information, facilitating a straightforward path to the solution.