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Chapter 1: Problem 113

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The product of \(\frac{2}{3}\) and a number, increased by \(6,\) is 3 less than the number.

Short Answer

Expert verified
The algebraic translation of the given English description is \(2x + 18 = 3x - 9\).

Step by step solution

01

Identify the Operations and Expressions

First, let's recognize all keyword phrases: 'The product of \(\frac{2}{3}\) and a number' means multiplication; 'increased by \(6\)' means addition; '3 less than the number' means subtraction.
02

Translate into an Equation

Secondly, let's translate this into an equation. 'The product of \(\frac{2}{3}\) and a number' can be written as \(\frac{2}{3} \cdot x\). 'increased by \(6\)' translates to \(+6\). 'Is' represents an '='. '3 less than the number' will be \(x-3\). Combine these to form: \(\frac{2}{3} \cdot x + 6 = x - 3\)
03

Simplify the Equation

Finally, we can simplify this equation by multiplying each term by \(3\) to get rid of the fraction, giving us \(2x + 18 = 3x - 9\).

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