Problem 37 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. one less than the product of 12 and a number

Step-by-Step Solution

Answer

The algebraic expression for the phrase is \(12x - 1\)

Step 1: Identify the variable

The problem mentions that 'x' represents the number.

Step 2: Recognize 'product'

The word 'product' indicates multiplication. So 'product of 12 and a number' translates to '12 times x' or '12x' in algebraic terms.

Step 3: Recognize 'less than'

The phrase 'less than' indicates subtraction. However, 'less than' is a little tricky because it means that the subtraction has to be done in reverse order. So 'one less than the product of 12 and a number' translates to 'subtract 1 from 12x', that's '12x - 1'.

Key Concepts

Variables

In algebra, variables are fundamental components. They are symbols used to represent unknown values or numbers. Think of them as placeholders. In most cases, variables are denoted by letters like \( x, y, \) or \( z \).
In our exercise, the variable \( x \) is used to denote a number that we need to work with. Variables allow us to create general expressions and equations that can be solved to find those unknown values. They can vary and can be substituted by different numbers in an equation, which gives equations their dynamic nature.

Multiplication in Algebra

Multiplication in algebra is a key operation and is often indicated by the term 'product'. Whenever you see the word 'product', it signifies a multiplication process.
This concept helps simplify expressions by condensing multiplication operations into concise terms such as \( 12x \). In this format, \( 12 \) is the coefficient, representing how many times the variable \( x \) is included in the calculation.
The ease of multiplying algebraic terms is crucial for simplifying complex expressions and solving equations. Remember, when dealing with algebraic multiplication:

• The order in which terms are multiplied does not matter, as multiplication is commutative.
• Coefficients are multiplied normally, while variables retain their initial forms unless specified otherwise.

Subtraction in Algebra

Subtraction in algebra helps to express how a quantity is diminished by another. The word 'less than' often signals subtraction, but be cautious about order. It's usually reversed.
For instance, 'one less than a product' implies we take 1 away from that product, not the other way around. So, in the expression '12x - 1', we start with \( 12x \) and subtract 1.
Some points to remember when subtracting in algebra:

• Ensure the order of terms reflects what the phrase indicates. 'Less than' means the number is subtracted from the expression or term presented first.
• Be consistent with practicing and recognizing subtraction vocabulary in various contexts to solidify understanding.

These techniques are essential as you encounter more complex problems.