Problem 48 - Elementary and Intermediate Algebra

Question

Translate to an algebraic expression. The product of 9 and twice \(m\)

Step-by-Step Solution

Answer

18m

Step 1: Understand the problem

Identify the key elements in the problem. The phrase 'product of 9 and twice m' means we need to multiply 9 by twice the variable 'm'.

Step 2: Translate 'twice m'

Twice a number means two times that number. Therefore, twice 'm' can be written as \(2m\).

Step 3: Express the entire phrase

Now, combine the elements. The product of 9 and \(2m\) means multiplying 9 by \(2m\). This can be written as \(9 \times 2m\).

Step 4: Simplify the expression

Simplify \(9 \times 2m\) to get the final algebraic expression, which is \(18m\).

Key Concepts

Translation to Algebra

When translating a word problem into algebra, start by identifying the essential elements in the problem. In our exercise, the key phrase is 'the product of 9 and twice m'.

Let’s break it down:

• 'Product' refers to multiplication.
• '9' is the constant.
• 'Twice m' means two times the variable 'm'.

This gives us the expression that combines these elements together. The goal is to write it in a form that you can use in algebraic operations.

For example, 'twice m' translates to '2m,' and the entire phrase 'the product of 9 and twice m,' translates to the multiplication of 9 and 2m, written as \(9 \times 2m\). Once translated, you’ll have the algebraic expression ready for further operations.

Multiplication in Algebra

Multiplication in algebra follows similar principles to arithmetic, but it also includes variables. Multiplying constants with variables involves straightforward steps. In the exercise, the phrase 'product of 9 and twice m' leads us to multiply 9 by 2m.

To multiply:

• First, multiply the constants (numbers) together.
• Then, include the variable with its coefficient.

Thus, multiplying 9 and 2m:
\[ 9 \times 2m = 18m \] Here, 9 and 2 are constants, and 'm' is the variable. Multiplying 9 by 2 gives 18, and we place 'm' alongside to get 18m.

This process simplifies the algebraic expression, making it ready for other operations such as addition, subtraction, or further multiplication.

Simplifying Expressions

Once we have translated and multiplied the components of our given problem, the final step in algebra involves simplifying the expression. Simplifying makes the expression easier to understand and use in further calculations.

From our exercise, the multiplication step provided us with: \[ 9 \times 2m = 18m \] This expression is already simplified because there are no like terms to combine or further operations to perform.

General principles for simplifying include:

• Combining like terms (terms that have the same variable and exponent).
• Performing arithmetic operations as needed.
• Ensuring that the expression is in its simplest form.

Simplifying expressions helps in solving equations effectively and makes the algebraic operations more manageable. An expression like 18m is ready-to-use and clearly represents the problem's conditions.