Problem 43 - Algebra A Combined Function

Question

Rewrite each sentence using mathematical symbols. Do not solve the equations. Twice a number, added to 6 , is 3 less than the number.

Step-by-Step Solution

Answer

Equation: \( 2x + 6 = x - 3 \)

Step 1: Identify the Unknown

The problem mentions a number, which we will represent with the variable \( x \).

Step 2: Translate Verbal Expressions into Algebraic Terms

The phrase 'twice a number' is translated into \( 2x \) since it means multiplying the number \( x \) by 2.

Step 3: Incorporate Additive Relationships

The phrase 'added to 6' means to add 6 to the expression, giving us \( 2x + 6 \).

Step 4: Set Up a Comparison

The phrase 'is 3 less than the number' translates to \( x - 3 \) since it involves subtracting 3 from the number \( x \).

Step 5: Combine the Expressions into an Equation

Translate the sentence into the equation \( 2x + 6 = x - 3 \) by equating the expressions from Steps 3 and 4.

Key Concepts

Variables and Unknowns

In algebra, variables and unknowns are fundamental concepts. A variable is a symbol—often a letter like \( x \)—that represents an unknown or changeable value within a mathematical expression. Recognizing variables is crucial in forming equations and understanding algebraic expressions.
\( x \) can change depending on the context of the problem, making it an unknown quantity at the start.
When given a problem like "Twice a number, added to 6, is 3 less than the number," the "number" becomes our variable. By letting \( x \) symbolize this unknown number, it helps in expressing the problem mathematically. Identifying what the variable stands for is the first step towards solving any algebraic equation.

Algebraic Translation

The ability to translate words into algebraic expressions is a key skill in algebra. This process involves interpreting verbal statements and converting them into mathematical symbols.
Consider the phrase "twice a number," which translates to \( 2x \). "Twice" suggests a multiplication by 2 of whatever the number is, represented by the variable \( x \).
Similarly, "added to 6" means taking \( 2x \) and adding 6 to it, forming the expression \( 2x + 6 \). Being able to accurately interpret each part of the verbal statement is essential for creating precise algebraic translations.

• "Twice a number" becomes \( 2x \).
• "Added to 6" translates to \( 2x + 6 \).
• "3 less than the number" is interpreted as \( x - 3 \).

Mastering this skill enables a smoother transition from words to equations, facilitating problem-solving in algebra.

Equation Formation

Forming an equation from a verbal description is the culmination of understanding variables and algebraic notation. It involves combining all parts of the problem into one complete mathematical sentence.
In our exercise, we identify "2x + 6" as one side of the equation, which comes from the translation "Twice a number, added to 6."
The other side, "3 less than the number" becomes \( x - 3 \).
To form the complete equation, set these two expressions equal to each other as they describe the same situation, yielding the equation \( 2x + 6 = x - 3 \).

• This represents the equality described in the problem.
• Each side of the equation corresponds to the expression derived from the verbal statement.
• Balancing both expressions ensures the accuracy of the equation.

Understanding equation formation is crucial, as it sets the stage for solving for the unknown and finding the solution.