Problem 55 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Find three consecutive odd integers whose sum is -213 .

Step-by-Step Solution

Answer

-73, -71, and -69

Step 1: Define the variables

Let the three consecutive odd integers be represented as \(x\), \(x+2\), and \(x+4\).

Step 2: Set up the equation

Use the information given in the problem to set up an equation. The sum of the three consecutive odd integers is -213. Thus, the equation is: \[ x + (x+2) + (x+4) = -213 \]

Step 3: Simplify the equation

Combine the like terms in the equation to simplify it: \[ 3x + 6 = -213 \]

Step 4: Solve for x

Isolate the variable by first subtracting 6 from both sides: \[ 3x + 6 - 6 = -213 - 6 \] \[ 3x = -219 \] Next, divide both sides by 3: \[ \frac{3x}{3} = \frac{-219}{3} \] \[ x = -73 \]

Step 5: Find the consecutive integers

Now use the value of \(x\) to find the three consecutive odd integers: \[ x = -73 \] \[ x + 2 = -73 + 2 = -71 \] \[ x + 4 = -73 + 4 = -69 \]

Step 6: Verify the solution

Check that the sum of the three integers equals -213: \[ -73 + (-71) + (-69) = -213 \] The solution is verified.

Key Concepts

consecutive odd integers

Consecutive odd integers are odd numbers that follow each other in sequence. For example, if you start with 3, the next consecutive odd integers are 5 and 7. These numbers are always two units apart because the sequence skips the even numbers in between.
To express consecutive odd integers algebraically, let the first odd integer be represented by a variable, say, \(x\). The next odd integer will then be \(x+2\), and the one after that will be \(x+4\). This pattern continues, adding 2 each time to get the next odd integer.
In the context of word problems, recognizing and defining consecutive odd integers clearly is essential. This helps in setting up the right equations to solve the problem efficiently.

solving equations

Solving equations involves finding the value of variables that make the equation true. Here's how we solve the problem involving consecutive odd integers' sum.
First, we identify the variables. Let's say the three consecutive odd integers are represented as \(x\), \(x+2\), and \(x+4\). Then, we set up an equation based on the given information.
The problem states that their sum is -213. So, we write: \[ x + (x+2) + (x+4) = -213 \]
Next, we simplify the equation by combining like terms: \[ 3x + 6 = -213 \]
To isolate \(x\), first, subtract 6 from both sides: \[ 3x + 6 - 6 = -213 - 6 \] \[ 3x = -219 \]
Then, divide both sides by 3: \[ \frac{3x}{3} = \frac{-219}{3} \] \[ x = -73 \]
Now, \(x\) is -73. Solving equations step-by-step ensures you accurately find the variable's value.

integer sums

Integer sums involve adding whole numbers, which can be positive, negative, or zero. Understanding integer sums is crucial for solving various algebra problems.
In our exercise, we calculate the sum of three consecutive odd integers. Once we solve for \(x\), we find the integers are \( -73, -71, \) and \( -69 \). We then verify the sum as follows:
First odd integer: \( -73 \)
Second odd integer: \( -71 \)
Third odd integer: \( -69 \)
The sum is: \[ -73 + (-71) + (-69) = -213 \]
Verification confirms the solution is correct.
Summing integers correctly, especially when involving negatives, is crucial for accurate problem-solving. Practice helps to master this skill and improves overall mathematical understanding.