Problem 96 - Algebra A Combined Function

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Eight added to twice a number is 42 .

Step-by-Step Solution

Answer

The equation is \(2x + 8 = 42\).

Step 1: Identify the unknown

The problem involves an unknown number, so we will use the variable \( x \) to represent it. Our task is to express the statement about this number mathematically.

Step 2: Translate verbal statement into mathematical terms

The statement "twice a number" is translated into \( 2x \), where \( x \) is the unknown number. The phrase "eight added to twice a number" means adding 8 to \( 2x \), giving us \( 2x + 8 \).

Step 3: Formulate the equation

The statement completes with "is 42," which indicates equality. Therefore, the mathematical equation that represents the problem is: \[ 2x + 8 = 42 \]

Key Concepts

Understanding an Unknown Variable

In algebra, an unknown variable is a symbol used to represent a number we don't know yet. It's like a placeholder that stands in for the unknown value in an equation. Usually, this variable is denoted by a letter like \( x \), \( y \), or \( z \). In our exercise, we want to find the number that, when used in a certain operation, results in a given total. Here, we use \( x \) as our symbol for the unknown number. Using variables helps us easily manipulate and solve mathematical expressions.

• Variables represent unknown values.
• Commonly denoted by letters like \( x \).
• Help in formulating and solving equations.

By learning to work with unknown variables, we gain the ability to explore and solve a wide range of problems, as variables provide flexibility in representing numbers that are not specified immediately.

Translating Verbal Statements

Translating verbal statements into mathematical language is a skill that can make problem-solving much simpler. It's about turning words into symbols and operations, which allows us to create equations that we can solve. For instance, the phrase "twice a number" translates to \( 2x \), indicating two times our unknown variable \( x \). This translation is crucial, as it anchors the process of putting the rest of the problem into an equation.

• Translate descriptive phrases to expressions.
• "Twice a number" becomes \( 2x \).
• Essential for forming precise equations.

By mastering this translation process, you can easily understand mathematical problems and express them in a way that makes it easier to find solutions.

Expressing Mathematical Terms

After translating verbal statements, the next step is to piece together a mathematical sentence. This often involves basic operations like addition, subtraction, multiplication, and division. In our exercise, "eight added to twice a number" becomes \( 2x + 8 \). Each part of the phrase must be carefully interpreted and written using mathematical terms to ensure it accurately reflects the problem.

• Use symbols to express operations: +, -, \( \times \), \( \div \).
• "Added to" means addition, as in \( 2x + 8 \).
• Accurate expression leads to correct problem solving.

When you're comfortable with expressing mathematical terms, you can create expressions and equations that precisely capture the intent of the original problem, making it easier to tackle complex mathematical tasks.

Forming Equations

Finally, forming an equation is about putting together all translated and expressed parts into a complete mathematical statement. An equation is a mathematical sentence that shows that two expressions are equal, marked by an "equals" sign \( = \). In our problem, the statement "is 42" completes the picture, resulting in the equation \( 2x + 8 = 42 \).

• Combine translated and expressed parts.
• Use the equals sign to show equality.
• In our case: \( 2x + 8 = 42 \).

Formulating equations from verbal problems allows us to use algebraic techniques to find the value of the unknown variable. By following these steps, you transform real-world situations into mathematical problems that you can solve using strategic methods.