Problem 15 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. 35 dollars more than twice the price

Step-by-Step Solution

Answer

The algebraic expression is \( 2x + 35 \).

Step 1: Identify the Unknown

The phrase involves an unknown quantity, which in this case, is 'the price'. Let's represent this unknown quantity with a variable, such as \( x \).

Step 2: Translate 'Twice the Price'

'Twice the price' means two times the price. Since we are using \( x \) to represent the price, twice the price is represented as \( 2x \).

Step 3: Translate '35 Dollars More'

The phrase '35 dollars more' indicates an addition of 35 to the existing quantity. We already have \( 2x \) and we need to add 35 to this value.

Step 4: Form the Algebraic Expression

Combine the parts to form the algebraic expression. '35 dollars more than twice the price' can be expressed as \( 2x + 35 \).

Key Concepts

Translating Phrases into Expressions

Understanding how to translate phrases into algebraic expressions is a fundamental skill in algebra. This process allows us to represent real-world situations with mathematical symbols, which can then be manipulated to solve problems. Consider the phrase '35 dollars more than twice the price'. To translate it into an expression, follow these steps:

• Identify Key Phrases: Look for words that indicate mathematical operations. In this example, 'twice the price' suggests multiplication, and '35 dollars more' suggests addition.
• Select a Variable: Decide on a letter to represent the unknown value. We'll use the variable \( x \) to represent 'the price'.
• Write the Expression: Translate each part of the phrase into mathematical terms. 'Twice the price' becomes \( 2x \), indicating two times \( x \), and '35 dollars more' adds 35, giving us \( 2x + 35 \).

By following these steps, you can effectively translate any verbal phrase into an algebraic expression.

Variables in Algebra

Variables play a crucial role in algebra. They serve as placeholders for numbers that can change or that we might not know yet. In our example, the variable \( x \) represents an unknown price.

Why Use Variables?

By using variables, we can work with unknown numbers in a flexible way. Think of them as blank spaces in a puzzle that need solving. This allows us to generalize and solve for multiple scenarios without changing the entire expression each time.

Choosing a Variable

When selecting a variable, flexibility is key. Typically, we use letters like \( x \), \( y \), or \( z \). However, any letter can be used, and sometimes, other symbols are appropriate depending on the context.Variables simplify complex problems by allowing us to write expressions that capture relationships between quantities. This makes them indispensable in solving algebraic problems.

Solving Algebraic Problems

Solving algebraic problems involves finding the value of unknown variables that satisfy given conditions. In our example, we translated a verbal phrase into the algebraic expression \( 2x + 35 \).

Steps to Solve Problems

• Set Up the Equation: Often, you'll translate a word problem into an equation. For instance, if you know the total cost must equal 75 dollars, you'd set \( 2x + 35 = 75 \).
• Solve for the Variable: Use algebraic operations to solve for the unknown variable. In our example, subtract 35 from both sides and then divide by 2 to find \( x \).
• Check Your Solution: After finding a solution, substitute it back into the original expression or equation to ensure it satisfies the conditions.

By transforming phrases into expressions and using variables, we can systematically approach and solve problems, making algebra a powerful tool in mathematics.