Problem 13 - Elementary Algebra Within Reach

Question

In Exercises \(11-22\), translate the verbal phrase into an algebraic expression. $$ b \text { decreased by } 25 $$

Step-by-Step Solution

Answer

\[ b - 25 \]

Step 1: Identify the variable

The problem gives us 'b'. This is our variable that represents a specific value.

Step 2: Identify the Operation required

'Decreased by' implicates subtraction. This means we have to subtract a certain value from our variable.

Step 3: Determine the Value to subtract

The phrase mentions that 'b' is decreased by 25. Therefore, 25 is the exact value that we subtract from 'b'.

Step 4: Formulate the Algebraic Expression

Put together what we have defined in the previous steps. We need to subtract 25 from 'b', which can be written as 'b - 25'.

Key Concepts

Algebraic Translation

Algebraic translation is a vital concept in math where words are transformed into mathematical expressions. When you come across a problem like "b decreased by 25," the phrase needs to be interpreted as an algebraic expression. This process is similar to learning a new language where each word or phrase has a corresponding mathematical symbol or operation.

• "Decreased by" is a keyword in this phrase that translates into the operation of subtraction.
• The word "b" in the phrase represents a variable which is an unknown value.

Understanding these translations helps to set up an equation that accurately represents the problem stated in words. Practice recognizing these key phrases to become more comfortable with translating verbal statements into algebraic expressions.

Variables

Variables are fundamental in algebra and play a crucial role in forming algebraic expressions. In the expression "b decreased by 25," the letter "b" represents the variable. A variable acts as a placeholder for an unknown value, allowing us to write flexible equations that can solve various problems.

• Variables are typically letters, like x, y, or in this case, b.
• They can take any value depending on the context of the problem.

Using variables makes it easier to generalize problems so they can be applied to different scenarios. Also, substituting different values for a variable helps find answers or test various conditions within a math problem.

Subtraction in Algebra

Subtraction in algebra is an operation that easily connects math to real-world scenarios, such as "b decreased by 25." Here, subtraction means you are taking away a certain number from another number or a variable. In our example, it involves subtracting 25 from "b."

• The expression becomes "b - 25," where 25 is the number that's subtracted.
• It's essential to remember that the order matters in subtraction—35 - 10 isn't the same as 10 - 35.

Understanding subtraction in algebra helps manage all sorts of numerical and real-life situations where you need to reduce a quantity by a specific amount. It's a straightforward operation yet highly significant when solving a wide range of algebra problems.