Problem 46 - 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 product of two numbers is twice their sum.

Step-by-Step Solution

Answer

\( xy = 2(x + y) \)

Step 1: Identify the unknowns

Let the two unknown numbers be represented by variables. Choose letters like x and y to represent these numbers.

Step 2: Express the product

The product of two numbers x and y can be represented by the expression: \( x \times y \) or \( xy \).

Step 3: Express the sum

The sum of the two numbers x and y is given by: \( x + y \).

Step 4: Express twice their sum

Twice the sum of the two numbers x and y is: \( 2(x + y) \).

Step 5: Formulate the equation

According to the sentence, 'The product of two numbers is twice their sum,' we set up the equation: \( xy = 2(x + y) \).

Key Concepts

algebraic expressions

An algebraic expression is a combination of numbers, variables, and mathematical operators like addition or multiplication. For example, in the sentence 'The product of two numbers is twice their sum,' we use algebraic expressions to represent the product and the sum of two unknown numbers.
Expressions like \( x \times y \) (product of x and y) and \( x + y \) (sum of x and y) are algebraic expressions.
They help us convert words into mathematical language, making it possible to solve problems using algebra.

equations

An equation states that two expressions are equal.
In our exercise, the sentence 'The product of two numbers is twice their sum' translates into an equation.
We have two expressions: the product \( xy \) and twice the sum \( 2(x + y) \).
By setting them equal, we get the equation: \( xy = 2(x + y) \).
This equation shows the relationship between the product and twice the sum of the two numbers.
Solving equations allows us to find the values of unknown variables that satisfy the given relationship.

variables

Variables are symbols that represent unknown values in algebraic expressions and equations.
In the exercise, we used the variables x and y to represent the two unknown numbers.
Variables provide a way to generalize problems and solve them using algebra. They can take any value, and by solving equations, we determine the specific values that satisfy the given conditions.
Understanding and using variables is essential for solving algebraic problems and translating sentences into equations effectively.