Problem 71 - Elementary and Intermediate Algebra

Question

In each of Exercises \(71-78,\) match the phrase or sentence with the appropriate expression or equation from the column on the right. Twice the sum of two numbers_____ a) \(\frac{x}{y}+6\) b) \(2(x+y)=48\) c) \(\frac{1}{2} \cdot a \cdot b\) d) \(t+2=5\) e) \(a b-1=49\) f) \(2(m+n)\) g) \(3(t+5)\) h) \(x-5=9\)

Step-by-Step Solution

Answer

f) 2(m+n)

Step 1: Identify key components of the phrase

The phrase 'Twice the sum of two numbers' can be broken down into 'twice' and 'the sum of two numbers'.

Step 2: Interpret 'Twice' mathematically

'Twice' means multiplying by 2. So we need to multiply the sum of two numbers by 2.

Step 3: Interpret 'Sum of two numbers' mathematically

If the two numbers are represented as variables, let's use m and n. The sum of m and n is simply written as (m+n).

Step 4: Form the expression

Combining the two interpretations, we get 2 times (m+n), which is written as 2(m+n).

Step 5: Match with given options

Among the given options, f) 2(m+n) matches our derived expression.

Key Concepts

Sum of Numbers in Algebra

When we talk about the sum of numbers, it means we are adding two or more numbers together. In algebra, these numbers are often represented by variables.
For instance, if we have two numbers represented as variables m and n, their sum is written as \(m + n\).
This is an essential concept in algebra that helps to create more complex expressions and equations.
Understanding how to add numbers is the foundation for many algebraic operations, making it easier to solve more complicated problems.
When breaking down phrases involving sums and other operations, look for keywords like 'sum,' 'total,' or 'together.' These words help identify that addition is necessary.

Multiplication in Algebra

Multiplication in algebra involves taking a number and increasing it repeatedly. In equations, it's often shown with variables.
For example, if we have the expression \(2 \cdot (m+n)\), this means we are multiplying the sum of m and n by 2.
This can be read as 'twice the sum of m and n.'
Multiplication is one of the basic operations you need to master to solve algebraic problems efficiently.

Keyword tips:

• Words like 'twice,' 'double,' or 'times' often indicate multiplication.
• Numbers being multiplied are called factors, and the result is the product.
• Sometimes multiplication is implied, such as in expressions like 2(m+n) without an explicit multiplication sign.

Matching Equations to Phrases

Matching equations to phrases helps bridge the gap between word problems and mathematical expressions.
This process involves understanding the language of math. For the given exercise, we matched 'Twice the sum of two numbers' to \(2(m+n)\).

Steps to match equations:

• Identify key terms: Break the phrase down into understandable parts. For instance, 'twice' suggests multiplication by 2.
• Interpret components: Understand that 'sum of two numbers' indicates addition, like \(m + n\).
• Combine parts: Put together your interpretations to form the expression, resulting in \(2(m+n)\).
• Match options: Compare your mathematical expression with the provided options to find a match.

This method works for various problems and helps solidify your understanding of algebraic concepts.