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Chapter 3: Problem 42

Translate each problem situation to a system of equations. Do not attempt to solve, but save for later use. The sum of two numbers is 30 . The first number is twice the second number. What are the numbers?

Short Answer

Expert verified
The system of equations is: \( \begin{cases} x + y = 30 \ x = 2y \end{cases} \).

Step by step solution

01

- Define the variables

Let the first number be denoted by \( x \) and the second number be denoted by \( y \).
02

- Write the sum equation

The problem states that the sum of the two numbers is 30. This can be represented by the equation \( x + y = 30 \).
03

- Write the relationship equation

The first number is twice the second number. This relationship can be expressed by the equation \( x = 2y \).
04

- Write the system of equations

Combining both equations, the system of equations is: \( \begin{cases} x + y = 30 \ x = 2y \end{cases} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Understanding algebraic expressions is crucial for solving the problems like the one given. An algebraic expression is a combination of variables, numbers, and operations. In our exercise, expressions like \( x + y \) and \( 2y \) play an important role.
By using algebraic expressions:
  • We can translate words into mathematical symbols. For instance, 'the sum of two numbers' becomes \( x + y \).
  • We can also express relationships. Here, 'the first number is twice the second number' becomes \( x = 2y \).
Learning to identify and write these expressions helps in setting up equations to solve later.
Defining Variables
Before forming equations, we must define the variables! This step means deciding what each variable represents.
In this problem:
  • We choose \( x \) to represent the first number.
  • We choose \( y \) to represent the second number.
Defining variables helps us to convert the word problem into a mathematical form. It simplifies understanding and aids in forming equations correctly.
Linear Equations
Linear equations are equations of the first degree, meaning they have no exponents higher than one. They are essential in forming a system of equations.
In our exercise, we have two linear equations:
  • \( x + y = 30 \), representing the sum of the two numbers.
  • \( x = 2y \), representing the relationship between the numbers.
Combining them forms a system of equations:\begin{cases} \ x + y = 30 \ x = 2y \ \end{cases}These equations help us understand the relationship between numbers and prepare us to solve for the unknowns later.

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Most popular questions from this chapter

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