Problem 18 - Algebra for College Students

Question

Translate each verbal sentence into an equation, using \(x\) as the variable. Then solve the equation. If the quotient of a number and 6 is added to twice the number, the result is 8 less than the number. Find the number.

Step-by-Step Solution

Answer

The number is \( \frac{-48}{7}\).

Step 1: Translate the verbal sentence into an equation

We need to convert the given verbal sentence into a mathematical equation. Let's break down the sentence: 'If the quotient of a number and 6 is added to twice the number, the result is 8 less than the number.' We'll use the variable \(x\) to represent the unknown number. The sentence can be translated as follows: 1. 'The quotient of a number and 6' translates to \(\frac{x}{6}\). 2. 'Twice the number' translates to \(2x\). 3. '8 less than the number' means \(x - 8\).Putting these parts together, the equation is: \[\frac{x}{6} + 2x = x - 8\]

Step 2: Simplify the equation

First, combine like terms and move all terms involving \(x\) to one side of the equation. Subtract \(x\) from both sides:\[\frac{x}{6} + 2x - x = -8\]Simplify the left side of the equation:\[\frac{x}{6} + x = -8\]

Step 3: Eliminate the fraction

To simplify further, eliminate the fraction by multiplying every term by 6, the denominator of the fraction: \[(6 \cdot \frac{x}{6}) + (6 \cdot x) = (6 \cdot -8) \ x + 6x = -48 \ 7x = -48 \ \]

Step 4: Solve for \( x \)

Now, solve for \( x \) by dividing both sides by 7:\[ x = \frac{-48}{7} \]

Key Concepts

Equation Solving

When we're translating a verbal sentence into an equation, our goal is to find a solution that satisfies the given conditions. Here, we started with the sentence: 'If the quotient of a number and 6 is added to twice the number, the result is 8 less than the number.' By breaking it down, we created an equation and then solved it step-by-step.

Variable Representation

In algebra, we often use letters to represent unknown values. These letters are called variables. In this exercise, we've used the variable 'x' to represent the unknown number. This helps simplify the sentence into a mathematical equation. Instead of saying 'a number,' we say 'x,' making it easier to handle mathematically.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. In translating the verbal sentence, we created expressions like \(\frac{x}{6}\), \(2x\), and \(x - 8\). These expressions allow us to form an equation that we can solve to find the value of 'x.' Using equations helps us understand relationships between different quantities and solve for unknowns.