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Chapter 2: Problem 5

The product of \(8,\) and a number increased by \(6,\) is \(104 .\) What is the number?

Short Answer

Expert verified
The number is 7.

Step by step solution

01

- Understand the Problem

The problem states that the product of 8 and a number, increased by 6, equals 104. Represent the number as a variable, say \(x\). The equation thus is \(8(x + 6) = 104\).
02

- Distribute the 8

Expand the equation by distributing the 8: \(8 \times x + 8 \times 6 = 104\). This simplifies to \(8x + 48 = 104\).
03

- Isolate the Variable

Subtract 48 from both sides to isolate the term with the variable: \(8x = 104 - 48\). This simplifies to \(8x = 56\).
04

- Solve for the Number

Divide both sides by 8 to solve for \(x\): \(x = \frac{56}{8}\). This simplifies to \(x = 7\).

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Key Concepts

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

linear equations
A linear equation is an equation in which the highest power of the variable is one. These equations can have one or more variables, but each variable will only have a power of one.
In the exercise, the equation is set up as follows: \(8(x + 6) = 104\). Here, we treat the variable \(x\) as a linear term because its highest power is one.
Linear equations are important because they are the simplest form of equations and form the basis for more complex algebra. When solving such equations, the objective is to find the value of the variable.
distributive property
The distributive property allows you to multiply a term outside the parentheses by each term inside the parentheses. In simpler words, it helps you 'distribute' the multiplier. This is written as: \(a(b + c) = ab + ac\).
In the given problem, we need to apply this property to simplify the equation \(8(x + 6)\). Using the distributive property, we rewrite it as \(8 \times x + 8 \times 6 = 104\), which further simplifies to \(8x + 48 = 104\).
This property is a crucial step in solving equations because it breaks down more complex expressions into simpler, manageable parts.
isolating variables
Isolating the variable is the process of rearranging an equation so that the variable stands alone on one side. This typically involves performing inverse operations to both sides of the equation.
For our equation \(8x + 48 = 104\), we first isolate the term with the variable by subtracting 48 from both sides: \(8x = 104 - 48\), which simplifies to \(8x = 56\).
The final step to fully isolate \(x\) is to divide both sides by 8, giving us \(x = \frac{56}{8}\), which simplifies to \(x = 7\).
Isolating the variable helps us find the solution to the equation, making it clear what value the variable must take to satisfy the equation.

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