Problem 51 - Algebra Form and Function

Question

The number of books you can afford to buy, \(b\), is a function of the number of \(\mathrm{CDs}, c,\) you buy and is given by \(b=10-0.5 c\). Which of the following equivalent expressions for this function most clearly shows the number of books you can afford if you buy \(6 \mathrm{CDs} ?\) (i) \(b=10-0.5 c\) (ii) \(\quad b=6-0.5(c-8)\) (iii) \(\quad b=7-0.5(c-6)\)

Step-by-Step Solution

Answer

Answer: The equivalent expression that most clearly shows the number of books you can afford if you buy 6 CDs is \(b=7-0.5(c-6)\).

Step 1: Analyze each expression

We will now analyze each expression by plugging in the value of \(c=6\) and see which one gives the simplest result.

Step 2: Analyze expression (i)

For expression (i), \(b=10-0.5c\), we plug in \(c=6\): $$ b=10-0.5(6)=10-3=7 $$

Step 3: Analyze expression (ii)

For expression (ii), \(b=6-0.5(c-8)\), we plug in \(c=6\): $$ b=6-0.5(6-8)=6-0.5(-2)=6+1=7 $$

Step 4: Analyze expression (iii)

For expression (iii), \(b=7-0.5(c-6)\), we plug in \(c=6\): $$ b=7-0.5(6-6)=7-0.5(0)=7 $$

Step 5: Compare the expressions

All three expressions give us the correct answer that we can afford \(7\) books when buying \(6\) CDs. However, expression (iii) \(b=7-0.5(c-6)\) most clearly shows this, as plugging in \(c=6\) directly gives the answer \(b=7\) without any additional calculations. Therefore, the equivalent expression that most clearly shows the number of books you can afford if you buy \(6\) CDs is: $$ b=7-0.5(c-6) $$

Key Concepts

Linear Equations

Linear equations are foundational in algebra and involve finding the values of unknown variables that satisfy a given equation. A linear equation takes the form of \( ax + b = 0 \) where \( a \) and \( b \) are constants, and \( x \) is the variable. These equations graph as straight lines when plotted on a coordinate plane.

Linear equations are vital because they often resemble real-world phenomena. In our exercise, the linear equation \( b = 10 - 0.5c \) signifies the relationship between two variables: the number of books you can afford \( b \) and the number of CDs \( c \). This relationship shows us how buying CDs decreases the number of books we can buy.

Remember, with linear equations, the variables interact in a simple, predictable way. As you increase \( c \) linearly, \( b \) decreases linearly as well. This predictability makes them incredibly useful in various scenarios from budgeting to planning. In essence, understanding linear equations helps interpret and model a multitude of situations.

Function Notation

Function notation streamlines the way we describe relations between different quantities using expressions. In mathematics, a function is a rule that assigns every input exactly one output. With function notation, we represent this as \( f(x) \) where \( x \) is the input of the function.

The exercise given uses function notation effectively by defining \( b = f(c) = 10 - 0.5c \). Here, \( b \) is expressed in terms of \( c \), showing a clear relationship where \( b \) varies according to changes in \( c \). This notation allows clarity and precision which is significant in mathematical communication.

Using function notation simplifies complex expressions and makes them manageable. For instance, it becomes easier to substitute and evaluate expressions – like determining the number of books for a specific number of CDs (e.g., when \( c = 6 \)). Employing function notation elevates our understanding of the relationship between quantities.

Solving Equations

Solving equations is the process of finding all possible values for the variables that make an equation true. This is a critical skill in algebra because it allows us to work through mathematical relationships to find unknown quantities.

In the given task, solving the equation involves substituting the value for \( c \) into the function to find \( b \). Each expression is examined by replacing \( c \) with \( 6 \) to see how many books can be afforded. For expression (i), the solution is straightforward: \( b = 10 - 0.5 \times 6 = 7 \). It checks if the equation holds true for specific conditions.

Understanding how to manipulate and simplify equations directly impacts our ability to solve real-world problems. Whether we're calculating budgets or predicting behavior outcomes, mastering equation-solving techniques is crucial for logical and analytical thinking. This exercise illustrates that solving equtions involves identifying the simplest and clearest expression that meets the problem's requirements.