Represent the unknown variable

Oct 13, 2025 by miodeya.com 47 views

Understanding the Problem

In this article, we will delve into the world of algebra and explore how to translate verbal phrases into algebraic expressions. We will take a specific phrase, "the square of a number increased by 9," and break it down into a mathematical expression. Then, we will evaluate the expression to find the value of the unknown variable.

Translating the Phrase

The phrase "the square of a number increased by 9" can be translated into an algebraic expression using the following steps:

Identify the key words: The key words in the phrase are "square," "number," and "increased by 9."
Determine the operation: The phrase indicates that we need to find the square of a number, which means we need to multiply the number by itself. The phrase also mentions that the result is increased by 9, which means we need to add 9 to the result.
Represent the unknown variable: Let's represent the unknown number as "n."
Write the algebraic expression: Based on the steps above, the algebraic expression for the phrase is:

n^2 + 9

This expression represents the square of the number (n^2) increased by 9.

Evaluating the Expression

Now that we have the algebraic expression, let's evaluate it to find the value of the unknown variable. However, we need to be given a value for the expression to solve for n. Since we are given multiple options, we will evaluate each option to see which one is correct.

Option A: $2n - 9 = 1$

This option is not related to the original phrase, so we can disregard it.

Option B: $n^2 - 9 = 16$

Let's evaluate this option by adding 9 to both sides of the equation:

n^2 = 16 + 9 n^2 = 25

Now, let's take the square root of both sides:

n = ±√25 n = ±5

This option is not correct because the original phrase does not indicate that the number is negative.

Option C: $2n + 9 = 19$

This option is not related to the original phrase, so we can disregard it.

Option D: $n^2 + 9 = 34$

Let's evaluate this option by subtracting 9 from both sides of the equation:

n^2 = 34 - 9 n^2 = 25

Now, let's take the square root of both sides:

n = ±√25 n = ±5

This option is correct because the original phrase indicates that the square of the number is increased by 9, and the result is equal to 34.

Conclusion

In this article, we translated the verbal phrase "the square of a number increased by 9" into an algebraic expression (n^2 + 9). We then evaluated the expression by using the given options to find the value of the unknown variable. The correct option is D: $n^2 + 9 = 34$ . This exercise demonstrates the importance of understanding verbal phrases and translating them into algebraic expressions to solve mathematical problems.

Key Takeaways

Verbal phrases can be translated into algebraic expressions using key words and operations.
The square of a number can be represented as n^2.
Adding 9 to the square of a number can be represented as n^2 + 9.
Evaluating an algebraic expression requires using the given information to solve for the unknown variable.

Practice Problems

Translate the verbal phrase "the cube of a number decreased by 7" into an algebraic expression.
Evaluate the algebraic expression 2x^2 - 5 to find the value of x.
Translate the verbal phrase "the product of two numbers increased by 3" into an algebraic expression.

References

Algebra: A Comprehensive Introduction
Mathematics for Dummies
Algebraic Expressions and Equations

Glossary

Algebraic expression: A mathematical expression that contains variables and constants.
Verbal phrase: A phrase that describes a mathematical operation or relationship.
Square: The result of multiplying a number by itself.
Cube: The result of multiplying a number by itself three times.
Frequently Asked Questions: Translating Verbal Phrases into Algebraic Expressions ====================================================================================

Q: What is the difference between a verbal phrase and an algebraic expression?

A: A verbal phrase is a phrase that describes a mathematical operation or relationship, while an algebraic expression is a mathematical expression that contains variables and constants.

Q: How do I translate a verbal phrase into an algebraic expression?

A: To translate a verbal phrase into an algebraic expression, follow these steps:

Identify the key words in the phrase.
Determine the operation indicated by the phrase.
Represent the unknown variable using a letter (e.g., x, y, n).
Write the algebraic expression using the key words and the unknown variable.

Q: What are some common verbal phrases that can be translated into algebraic expressions?

A: Some common verbal phrases that can be translated into algebraic expressions include:

"The square of a number" (n^2)
"The cube of a number" (n^3)
"The product of two numbers" (xy)
"The sum of two numbers" (x + y)
"The difference of two numbers" (x - y)

Q: How do I evaluate an algebraic expression?

A: To evaluate an algebraic expression, follow these steps:

Simplify the expression by combining like terms.
Use the order of operations (PEMDAS) to evaluate the expression.
Substitute the given values into the expression.
Solve for the unknown variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

Combine like terms by adding or subtracting coefficients.
Eliminate any unnecessary parentheses or brackets.
Use the distributive property to expand expressions.
Simplify any exponential expressions.

Q: What are some common algebraic expressions that can be simplified?

A: Some common algebraic expressions that can be simplified include:

2x + 3x = 5x
x^2 + 4x + 4 = (x + 2)^2
3x^2 - 2x^2 = x^2

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, follow these steps:

Simplify the equation by combining like terms.
Use the order of operations (PEMDAS) to evaluate the equation.
Isolate the unknown variable by adding, subtracting, multiplying, or dividing both sides of the equation.
Check the solution by substituting it back into the original equation.

Q: What are some common algebraic equations that can be solved?

A: Some common algebraic equations that can be solved include:

2x + 3 = 5
x^2 + 4x + 4 = 0
3x^2 - 2x^2 = x^2

Conclusion

In this article, we have covered some of the most frequently asked questions about translating verbal phrases into algebraic expressions. We have also discussed how to evaluate, simplify, and solve algebraic expressions and equations. By following these steps and practicing with examples, you can become proficient in translating verbal phrases into algebraic expressions and solving algebraic equations.