Problem 69 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression. Use \(x\) for the unknown number. The sum of -3 and twice a number

Step-by-Step Solution

Answer

The expression is \(-3 + 2x\).

Step 1: Identify the Components

First, identify the elements in the phrase: 'the sum of -3 and twice a number'. The two main components are '-3' and 'twice a number'. 'Twice a number' implies multiplying the unknown number by 2, and we are instructed to use \( x \) for the unknown number.

Step 2: Formulate the Expression for 'Twice a Number'

Since 'twice a number' means multiplying the number by 2, represent this part as \( 2x \). This expression reflects 'twice the unknown number \( x \)'.

Step 3: Combine for the Sum

The phrase asks for the sum of -3 and \( 2x \). In mathematical terms, 'sum' means you add the numbers together. Therefore, write the algebraic expression as \(-3 + 2x\).

Key Concepts

Unknown Variable

In algebra, an unknown variable is often used to represent a number that we do not yet know. Variables are usually denoted by letters such as \( x \), \( y \), or \( z \). In our exercise, we are asked to use \( x \) as the unknown number. This means that wherever there is a reference to the unknown number, we will mathematically express it with \( x \).

Using variables allows us to write expressions and equations that can be solved to find the value of the unknown. They act like placeholders and make it easier to work with numbers that change or are undetermined. By treating \( x \) as the number we need to find, we can easily manipulate the expression mathematically.

Sum in Algebra

In algebra, the term "sum" indicates the addition of two or more quantities. When you see 'the sum of' in a phrase, it means you need to add up the components involved.

For the exercise given, the phrase 'the sum of -3 and twice a number' tells us to add \(-3\) to another component, which is twice the unknown number.

To express this mathematically, we look for the two elements that need to be summed up. Here, they are \(-3\) and \(2x\). Adding them up gives us the expression:

• \(-3 + 2x\)

Algebraic sums may involve constants, coefficients, and variables, allowing us to represent real-world scenarios mathematically.

Multiplication in Algebra

Multiplication in algebra involves combining a number (coefficient) and a variable to produce a new term. For instance, 'twice a number' refers to multiplying two with that number.

In our exercise, 'twice a number' translates to \(2 \times x\), which is straightforwardly written as \(2x\). The multiplication here is implied, as often in algebra, there is no need to use a multiplication sign between numbers and variables.

This simplification allows us to concisely express ideas. Multiplication of variables by coefficients is a fundamental operation in creating algebraic expressions, which is critical in solving equations and modeling situations around us.