Problem 44 - Elementary and Intermediate Algebra

Question

To prepare for Section 9.2, review translating sentences to equations (Section 1.1). Translate each sentence to an equation.\([1.1]\) The sum of three numbers is \(100 .\)

Step-by-Step Solution

Answer

The equation is \(x + y + z = 100\).

Step 1: Understand the Sentence

Identify the key components in the sentence. Here, we have three numbers, their sum, and the total sum being 100.

Step 2: Define Variables

Let's assign variables to the three numbers. Let the three numbers be represented by the variables: \(x\), \(y\), and \(z\).

Step 3: Form the Equation

Write down the equation representing the sum of the three numbers: \(x + y + z = 100\).

Key Concepts

understanding sentences in algebra

Understanding sentences in algebra is about turning everyday language into mathematical language. For instance, consider the sentence: 'The sum of three numbers is 100.' To begin, identify keywords and phrases that indicate mathematical operations. Here, 'sum' suggests an addition operation, and 'is' indicates equality. The phrase 'three numbers' implies we are adding three different values together. Breaking down this sentence helps us see that we need to find three numbers that add up to 100. Recognizing these keywords and their respective operations is the first step in translating sentences to equations.
Now that we understand how to break down the sentence, we can proceed to defining our variables.

defining variables

Defining variables is a key step in forming equations. Variables serve as placeholders for unknown values. In our sentence, we need to represent three unknown numbers. We can use letters to represent these numbers, such as:

• x
• y
• z

Using these variables, we can easily refer to each unknown number without confusion. It's important to choose variables that are easy to remember and distinguish from one another.
In this case, we'll let the three numbers be represented by the variables:

• \( x \)
• \( y \)
• \( z \)

Each variable stands for one of the unknown numbers whose sum is 100.
This step prepares us for the final stage where we form the actual equation based on our defined variables.

forming equations

Forming equations involves using our understanding of the sentence and the defined variables to write a mathematical statement. In our example, we are told that the sum of three numbers is 100. With our defined variables: \( x \), \( y \), and \( z \), we can now construct the equation.
Since 'sum' implies addition, we will add the three variables together to equal 100: \[ x + y + z = 100 \] This equation now precisely captures the information provided in the original sentence. It tells us that when the values of \( x \), \( y \), and \( z \) are added together, the result is 100. Forming equations in this way allows us to solve for unknown values, verify information, or make predictions using algebra.
By translating the sentence into a mathematical equation, we have completed the process of turning words into an actionable algebraic form.