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Chapter 3: Problem 67

One number is 5 less than 4 times another. If the sum of the two numbers is 11 find the numbers.

Short Answer

Expert verified
The numbers are \( \frac{16}{5} \) and \( \frac{39}{5} \).

Step by step solution

01

Define Variables

Let the first number be represented by \( x \). Since one number is 5 less than 4 times another, the second number can be represented as \( 4x - 5 \).
02

Set Up Equation

The sum of the two numbers is 11. Therefore, we set up the following equation: \( x + (4x - 5) = 11 \).
03

Simplify the Equation

Combine like terms in the equation: \( x + 4x - 5 = 11 \). This simplifies to \( 5x - 5 = 11 \).
04

Solve for \( x \)

Add 5 to both sides of the equation to isolate the \( 5x \) term: \( 5x - 5 + 5 = 11 + 5 \). This simplifies to \( 5x = 16 \). Then, divide both sides by 5: \( x = \frac{16}{5} \).
05

Find the Second Number

Substitute \( x = \frac{16}{5} \) back into the expression for the second number: \( 4x - 5 \). This gives \( 4(\frac{16}{5}) - 5 = \frac{64}{5} - \frac{25}{5} = \frac{39}{5} \).

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Key Concepts

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

Variable Definition
Defining variables is the first and crucial step in solving linear equations. Variables act as placeholders for the unknown values we want to find.
In this exercise, we need to find two numbers. Let's call the first number \( x \). The second number is described as 5 less than 4 times the first number. Therefore, we represent the second number as \( 4x - 5 \). By defining variables, we can transform a word problem into a mathematical equation that is easier to work with.
Solving Equations
Once we've defined our variables, the next step is to set up and solve the equation.
The problem states that the sum of the two numbers is 11. This translates to:
\( x + (4x - 5) = 11 \).
Now, we have an equation to solve.
Solving an equation involves finding the value of the variable that makes the equation true.
To do this, the equation needs to be simplified and manipulated to isolate the variable.
Combining Like Terms
Combining like terms is an important step in simplifying equations.
Like terms are terms that have the same variable raised to the same power.
In our equation \( x + 4x - 5 = 11 \), we see two like terms: \( x \) and \( 4x \).
Combining them, we get:
\( 5x - 5 = 11 \)
This simplification makes it easier to solve the equation.
Simplification
Simplifying the equation involves performing basic arithmetic operations to isolate the variable.
From our combined terms, we have:
\( 5x - 5 = 11 \).
First, we add 5 to both sides to isolate the term with the variable:
\( 5x - 5 + 5 = 11 + 5 \).
Simplifying further, we get:
\( 5x = 16 \).
Next, we divide both sides by 5 to solve for \( x \):
\( x = \frac{16}{5} \).
To find the second number, substitute \( x \) back into the expression for the second number:
\( 4(\frac{16}{5}) - 5 \).
This results in \( \frac{64}{5} - \frac{25}{5} = \frac{39}{5} \).
Therefore, the two numbers are \( \frac{16}{5} \) and \( \frac{39}{5} \).

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Most popular questions from this chapter

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