Translate The Sentence Six Less Than The Product Of Four And A Number Is -2 Into An Equation

Translate The Sentence Six Less Than The Product Of Four And A Number Is -2 Into An Equation

When approaching word problems in mathematics, the key is to carefully interpret the language and convert it into an algebraic expression or equation. The sentence "Six less than the product of four and a number is -2" provides specific clues about the relationships between variables and constants. To translate this sentence into an equation, it's essential to understand the meaning of each phrase and how they connect mathematically. This article will guide you step-by-step through the process of translating such sentences into algebraic equations, with a focus on the given problem.

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Understanding the Components of the Sentence

Before translating the sentence into an equation, it's important to break down its parts and understand what each segment signifies. The sentence contains several key components:

Identifying the Unknown Variable

• The phrase "a number" indicates the presence of an unknown quantity, which we typically represent with a variable, usually \( x \).

Interpreting "the product of four and a number"

• The word "product" signifies multiplication.
• "Four and a number" suggests multiplying 4 by the unknown variable \( x \).

Understanding "Six less than"

• The phrase "less than" indicates subtraction.
• "Six less than" means subtracting 6 from something.

Recognizing the equality or relationship

• The phrase "is -2" shows that the expression on the left side equals -2.

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Step-by-Step Translation Process

Translating a word problem into an algebraic equation involves several systematic steps:

Step 1: Assign a Variable to the Unknown

• Choose a variable to represent "a number," commonly \( x \).

Step 2: Express the Product of Four and the Number

• Translate "the product of four and a number" into \( 4x \).

Step 3: Interpret "Six less than" the Product

• Since "six less than" indicates subtraction, express it as \( 4x - 6 \).

Step 4: Set the Expression Equal to -2

• The phrase "is -2" indicates an equality, so set the expression equal to -2:

\[ 4x - 6 = -2 \]

Step 5: Final Equation

• The complete translation results in the algebraic equation:

\[ 4x - 6 = -2 \]

This equation now accurately represents the original sentence, ready for solving.

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Understanding the Structure of the Equation

The equation \( 4x - 6 = -2 \) encapsulates the meaning of the sentence in algebraic form. Let's analyze its components:

Left Side: \( 4x - 6 \)

• Represents "six less than the product of four and a number."
• \( 4x \): product of 4 and the unknown number \( x \).
• Subtracting 6 aligns with "less than."

Right Side: \(-2\)

• The value to which the expression equals, as stated in the sentence.

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Additional Examples and Variations

Different word problems might have similar structures but vary slightly in wording. Here are some examples and their translations:

• Example 1: "Three more than twice a number equals ten."
• Translation: \( 2x + 3 = 10 \)
• Example 2: "The sum of a number and five is equal to twelve."
• Translation: \( x + 5 = 12 \)
• Example 3: "Five less than triple a number is twenty."
• Translation: \( 3x - 5 = 20 \)

Understanding the pattern of such sentences helps in systematically translating them into equations.

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Solving the Equation \( 4x - 6 = -2 \)

Once the translation is complete, the next step is to solve the equation to find the value of the unknown \( x \).

Step 1: Isolate the Variable Term

• Add 6 to both sides to eliminate the constant term on the left:

\[ 4x - 6 + 6 = -2 + 6 \] \[ 4x = 4 \]

Step 2: Solve for \( x \)

• Divide both sides by 4:

\[ \frac{4x}{4} = \frac{4}{4} \] \[ x = 1 \]

Solution: The number is \( x = 1 \).

Step 3: Verify the Solution

• Substitute \( x = 1 \) back into the original expression:

\[ 4(1) - 6 = 4 - 6 = -2 \]

• The left side equals the right side, confirming that the solution is correct.

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Importance of Accurate Translation in Word Problems

Translating words into equations is a fundamental skill in algebra. Accurate translation ensures correct problem-solving and understanding. Common mistakes include:

• Misinterpreting "less than" or "more than."
• Forgetting to set the expression equal to the correct value.
• Incorrectly translating "product," "sum," or "difference."

By developing a methodical approach—identifying key phrases, assigning variables, and translating step-by-step—you can effectively convert complex sentences into solvable equations.

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Conclusion

Translating the sentence "Six less than the product of four and a number is -2" into an algebraic equation involves understanding the language of the problem and carefully expressing each part mathematically. The key steps include:

• Assigning a variable to the unknown.


• Converting descriptive phrases into mathematical operations.


• Setting the resulting expression equal to the given value.

The final equation, \( 4x - 6 = -2 \), provides a clear pathway to solving for the unknown. Mastering this translation process enhances problem-solving skills and lays a foundation for tackling more complex algebraic word problems. Remember, clarity in interpretation leads to accuracy in solution.