Problem 47 - Algebra A Combined Function

Question

Rewrite each sentence using mathematical symbols. Do not solve the equations. The product of 4 and the sum of a number and 6 is twice the number.

Step-by-Step Solution

Answer

The equation is \(4(x + 6) = 2x\).

Step 1: Identify Variables and Operations

First, define the variable in the sentence. Let the unknown number be represented by the variable \( x \). Identify the key operations: multiplication (product of 4 and a sum) and addition (sum of a number and 6).

Step 2: Construct the Sum Expression

The sum of the unknown number \( x \) and 6 can be written as an expression: \( x + 6 \).

Step 3: Construct the Product Expression

The product of 4 and the expression \( x + 6 \) is written as: \( 4(x + 6) \).

Step 4: Translate 'Twice the Number'

The phrase "twice the number" means multiplying the number \( x \) by 2. This can be written as: \( 2x \).

Step 5: Write the Equation

Combine these expressions into an equation. The product of 4 and the sum of a number and 6 (\(4(x + 6)\)) equals twice the number (\(2x\)). The equation is: \[ 4(x + 6) = 2x \].

Key Concepts

Understanding Variables in Algebra

In algebra, variables are symbols that represent unknown values in a mathematical expression or equation. These symbols are generally letters of the alphabet, like \( x \), \( y \), or \( z \). They serve as placeholders for numbers that we are yet to determine.

• Defining Variables: In the given exercise, the letter \( x \) is used to stand for 'a number'. We don't know what this number is, which is why we use a variable to represent it.
• Why Use Variables: Using variables can help us model real-world situations where exact numbers are variable or unknown, enabling us to solve a wide range of problems.

Variables are essential for forming expressions, constructing equations, and finding solutions in algebra.

Constructing Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (adds, subtracts, multiplies, divides). Constructing expressions involves representing a word problem or situation mathematically.

• Addition Expressions: In the example exercise, 'the sum of a number and 6' is translated to \( x + 6 \).
• Multiplication Expressions: 'The product of 4 and the sum' translates to 4 times the expression, which is written as \( 4(x + 6) \).

Constructing expressions is vital as it transforms a verbal problem statement into a mathematical form, making it easier to analyze and solve.

Equations in Algebra

An algebraic equation is like a balance, depicting two equal expressions separated by an equals sign \( = \). It tells us that two mathematical expressions represent the same quantity.

• Understanding Equality: In the problem, the equation states that the product of 4 and the sum of a number and 6 is equal to twice the number, represented by \( 4(x + 6) = 2x \).
• Components of an Equation: On the left, we have \( 4(x + 6) \), which means we multiply 4 by \( x + 6 \). On the right, \( 2x \) implies doubling \( x \).

Equations help us work out what a variable stands for by setting different expressions equal to each other, which we then solve through algebraic manipulations.