Problem 28 - Algebra 1

Question

Write the verbal sentence as an equation or an inequality. Five decreased by eight is four times \(y.\)

Step-by-Step Solution

Answer

The equation is \(5 - 8 = 4y\).

Step 1: Interpretation

In the statement, 'Five decreased by eight', the phrase 'decreased by' signifies subtraction. Thus, this translates to '5 - 8'.

Step 2: Interpretation Continued

The next part of the sentence is 'is four times \(y\).' The word 'is' signifies 'equal to' in mathematical terms, and 'four times \(y\)' translates to '4y'.

Step 3: Form the equation

Combining the above interpretations, 'Five decreased by eight is four times \(y\)' simply translates to '5 - 8 = 4y'.

Key Concepts

Equations

Equations are like mathematical sentences that show the relationship between different quantities. They involve different components such as numbers, symbols, and operations (like addition or subtraction) that together express a complete idea. In an equation, the equal sign \( = \) is the key feature, showing that what is on the left side has the same value as what is on the right side.
In the exercise, we turned a verbal statement into an equation: "Five decreased by eight equals four times \( y \)." This becomes \( 5 - 8 = 4y \). This equation tells us that when you subtract 8 from 5, the result is the same as multiplying a variable \( y \) by 4.
When working with equations, knowing how to translate words into symbols accurately is important. This helps us understand and solve equations effectively.

Variables

Variables in mathematics are symbols, often lettered, that stand for unknown or changeable values. They give us a way to explore mathematical relationships without being tied to specific numbers. In the exercise, \( y \) is the variable. It represents a number we do not yet know but want to find or express in terms of other numbers.
By using variables, we can write and solve equations that model real-world situations. For example, in our problem, \( 4y \) means "four times whatever value \( y \) has." Using variables makes equations powerful tools for solving everyday problems and understanding patterns.
Remember, a variable can be any letter, and its meaning depends on how it appears in an equation.

Mathematical Translations

Mathematical translations are all about converting everyday language into mathematical expressions or equations. This skill is crucial because it allows us to think and solve problems mathematically. In the exercise, we translated a verbal sentence into a mathematical equation.
Here's how translations work:

• Identify key phrases: Words like "decreased by" mean subtraction, and "times" means multiplication.
• Recognize equality indicators: Words like "is" or "equals" translate to the equal sign \( = \).
• Associate words with operations: For instance, "five decreased by eight" turns into \( 5 - 8 \).

By mastering translations, you can take complex problems described in words and craft equations that are clear and ready to be solved. This bridges the gap between verbal descriptions and mathematical solutions, making it easier to understand and express mathematical ideas.