Problem 44 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two consecutive integers is \(-37 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are -19 and -18.

Step 1 - Understand the problem

Two consecutive integers are numbers that follow each other in order. If one integer is denoted as \( n \), the next consecutive integer is \( n + 1 \). The problem states that their sum is -37.

Step 2 - Set up the equation

Write an equation based on the information given: \[ n + (n + 1) = -37. \]

Step 3 - Combine like terms

Combine the \( n \) terms in the equation: \[ 2n + 1 = -37. \]

Step 4 - Isolate the variable \( n \)

Subtract 1 from both sides to isolate the term with \( n \): \[ 2n = -37 - 1, \] which simplifies to \[ 2n = -38. \]

Step 5 - Solve for \( n \)

Divide both sides by 2 to find \( n \): \[ n = \frac{-38}{2}, \] which simplifies to \[ n = -19. \]

Step 6 - Find the consecutive integer

The consecutive integer next to \( n = -19 \) is \( n + 1 \), so it is \[ -19 + 1 = -18. \]

Step 7 - Verify the solution

The sum of the integers is \[ -19 + (-18) = -37. \] This confirms the solution is correct.

Key Concepts

Number Word Problems

Number word problems often describe mathematical situations in words. These problems usually require forming and solving equations to find one or more unknown numbers. The key to solving such problems is to translate the words into algebraic expressions or equations. For example, in the provided problem, you are told to find two consecutive integers whose sum is -37. To convert this into a solvable problem, you can represent the integers as variables and set up an equation based on the given sum.

Integer Operations

Integer operations are fundamental in solving equations that involve whole numbers. Integers are positive and negative whole numbers, including zero. In the problem (-19) + (-18) = -37, some common operations include:

• Addition: Combining numbers to get their sum.
• Subtraction: Finding the difference between numbers.
• Multiplication: Scaling one number by another.
• Division: Splitting a number into equal parts.

Often, working with negative integers can be tricky. Remember that adding a negative number means moving to the left on a number line, while subtracting a negative number is like adding its positive counterpart.

Solving Equations

To solve equations, you need to isolate the variable (usually represented by letters like n or x) to find its value. Here’s a step-by-step approach:

• Set up the equation based on the problem statement.
• Combine like terms to simplify the equation.
• Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable.
• Solve for the variable.
• Verify the solution by substituting it back into the original equation.

For example, in the given exercise, you set up the equation as n + (n + 1) = -37 and simplify it step by step until you find the values of the consecutive integers. This method ensures that you systematically solve for the unknowns in a structured way.
Understanding these steps will help you tackle similar word problems effectively.