Problem 47 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Find three consecutive integers whose sum is -36 .

Step-by-Step Solution

Answer

The three consecutive integers are -13, -12, and -11.

Step 1 - Define the integers

Let the three consecutive integers be defined as follows: the first integer is denoted as \(x\), the second integer as \(x + 1\), and the third integer as \(x + 2\).

Step 2 - Set up the equation

Write an equation representing the sum of these three integers: x + (x + 1) + (x + 2) = -36.

Step 3 - Combine like terms

Combine the like terms on the left side of the equation: 3x + 3 = -36.

Step 4 - Isolate the variable

Subtract 3 from both sides to isolate the term with the variable: 3x + 3 - 3 = -36 - 3 3x = -39.

Step 5 - Solve for x

Divide both sides by 3 to find the value of \(x\): x = \frac{-39}{3} x = -13.

Step 6 - Find the consecutive integers

Having found \(x = -13\), substitute back to get the three consecutive integers: -13, -12, -11.

Key Concepts

Solving Equations

To solve equations, you need to uncover the value of the variable that makes the equation true.
Think of it as a balance scale, where both sides need to be equal. You'll perform operations like addition, subtraction, multiplication, or division to isolate the variable.

For our problem, the three consecutive integers sum up to -36. We start with forming an equation from the word problem and then simplify it step by step.

Here's a quick guide to solving equations:

• Combine like terms.
• Perform inverse operations to isolate the variable.
• Simplify the equation until the variable stands alone.

After defining the variables as three consecutive integers \(x\), \(x+1\), \(x+2\), we set up the equation as \(x + (x + 1) + (x + 2) = -36\) and follow these steps.

Consecutive Integers

Consecutive integers are numbers that follow each other in order, one after the other.
The pattern is simple and predictable, making them easier to work with in word problems.
For example, if you start with an integer \(x\), the next consecutive integer is \(x + 1\), and the one following that is \(x + 2\). So, if our starting number is -13, the consecutive sequence would be -13, -12, and -11.

Understanding this concept lets you quickly translate word problems into equations. When given a problem asking for 'consecutive integers', always remember they increase by one unit each time.

Integer Solutions

Integer solutions are whole numbers, both positive and negative, that solve an equation.
In many word problems, especially those involving sums or products, your final answer will often be an integer.

In our problem, we ended up with \(x = -13\), meaning our integers are -13, -12, and -11.

To check the solution, simply add these integers to verify they sum up to -36:

• \(-13 + (-12) + (-11) = -36\)

We have verified that our solution is correct. Always double-check your results to ensure your integer solutions are accurate and make sense in the context of the problem.