Problem 42 - 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 difference of two numbers is twice the first number.

Step-by-Step Solution

Answer

The equation is y - x = 2x.

Step 1: Identify the unknowns

Let's denote the two numbers as variables. Let the first number be represented by the variable 'x' and the second number be represented by the variable 'y'.

Step 2: Write the expression for the difference

According to the sentence, 'The difference of two numbers' can be mathematically expressed as the subtraction of 'x' from 'y', which can be written as y - x

Step 3: Translate 'twice the first number'

The sentence states that the difference is 'twice the first number'. 'Twice the first number' implies multiplying the first number by 2, which can be written as: 2x

Step 4: Set up the equation

According to the sentence, 'The difference of two numbers is twice the first number'. Hence, we can set up the equation: y - x = 2x

Key Concepts

Identifying Unknowns

When translating sentences into equations, the first crucial step is to identify the unknowns.
Unknowns are the values we don't know yet and need to figure out.
They are usually represented by variables like 'x', 'y', or 'z'.
In our example, we have two unknowns: the first number and the second number.
We can name these unknowns using variables:

• Let 'x' be the first number.
• Let 'y' be the second number.

Naming our unknowns with variables makes it easier to manipulate and solve equations.
It's essential to choose variable names that make sense for the context.
This ensures that our mathematical expressions are clear and straightforward to follow.

Writing Expressions

Once we've identified the unknowns, the next step is to write mathematical expressions based on the given sentences.
In our problem, the sentence states, 'The difference of two numbers.'
Here, 'difference' refers to subtraction.
Therefore, we can write this difference as:
\[ y - x \]
We also see the phrase 'twice the first number,' which translates to multiplying the first number by 2:
\[ 2x \]
By carefully translating each part of the sentence into a mathematical expression, we ensure no details are lost.
These expressions lay the groundwork for forming our equations.

Setting Up Equations

Setting up equations involves combining our mathematical expressions to form a complete equation that represents the given sentence.
In this exercise, the sentence states: 'The difference of two numbers is twice the first number.'
We've already expressed the 'difference of two numbers' as \[ y - x \].
We've also translated 'twice the first number' as \[ 2x \].
According to the problem, these two expressions are equal.
Hence, we set up the equation: \[ y - x = 2x \].
This equation now represents the entire sentence mathematically.
Setting up correct equations is crucial in solving mathematical problems effectively.
It ties together all parts of the sentence into a single, solvable mathematical statement.