Problem 18 - Algebra Form and Function

Question

The balance in a checking account set up to pay rent, \(m\) months after its establishment, is given by $$ 4800-\( \)400 m\( (a) Write an equation whose solution is the number of months it takes for the account balance to reach \) 2000\( (b) Make a plot of the balance for m=1,3,5,7,9,11 and indicate the solution \)m=7$ to the equation in part (a).

Step-by-Step Solution

Answer

Answer: It takes 7 months for the account balance to reach $2000. The balances for every 2 months are as follows: (1) $4400, (3) $3600, (5) $2800, and (7) $2000.

Step 1: 1. Write the Equation for Account Balance of \(2000

To find the number of months it takes for the account balance to reach \)2000, set the given equation, $$4800 - 400m$$, equal to $2000: \(4800 - 400m = 2000\)

Step 2: 2. Solve for the Number of Months, \(m\)

Now, we need to solve the equation for \(m\): $4800 - 400m = 2000 \\ -400m = -2800 \\ m = 7$

Step 3: 3. Create a Table of Values for the Balance

Using the given equation, we'll create a table of values for \(m=1,3,5,7,9,11\): | \(m\) | Balance | |-----|---------| | 1 | 4400 | | 3 | 3600 | | 5 | 2800 | | 7 | 2000 | | 9 | 1200 | |11 | 400 |

Step 4: 4. Plot the Points and Indicate the Solution \(m=7\)

Plot the points from the table on a graph: (1, 4400), (3, 3600), (5, 2800), (7, 2000), (9, 1200), and (11, 400) Highlight the point (7, 2000) as the solution to part (a), indicating that it takes 7 months for the account balance to reach $2000.

Key Concepts

Solving Equations

Solving equations is a process used to find the value of a variable that makes a mathematical equation true. In our exercise, we're dealing with the linear equation \(4800 - 400m = 2000\). This equation represents the balance of a checking account over time, specifically how many months it will take for the balance to reach \(2000.

To solve this equation, we follow these steps:

• Start by simplifying the equation, if necessary. Here, we need to isolate \(m\).
• Subtract 4800 from each side to get \(-400m = -2800\).
• Next, divide each side by -400 to solve for \(m\): \(m = 7\).

Therefore, the number of months it takes for the balance to reach \)2000 is 7 months. Solving linear equations like this one helps us understand relationships and changes over time.

Graphing

Graphing can visually represent mathematical relationships and solutions. By plotting points on a graph, we can gain insights into the behavior of a function and identify trends or solutions. In this exercise, graphing helps us see how the balance changes over time.

Firstly, convert your equation into points with their corresponding values. In our case:

• For \(m=1\): \(4400\)
• For \(m=3\): \(3600\)
• For \(m=5\): \(2800\)
• For \(m=7\): \(2000\)
• For \(m=9\): \(1200\)
• For \(m=11\): \(400\)

Plot these points on a graph where the x-axis represents months \(m\) and the y-axis represents balance. The resulting line will slope downward, indicating a consistent decrease in the balance over time.

Additionally, when graphing, emphasize the solution point found in the calculation: \(m=7\), with a balance of $2000. This point shows us the exact time when the balance reaches this amount.

Tables of Values

Tables of values are a great way to organize data and make complex information more digestible. By calculating specific balances at given months using the equation, we create a clear table, making the trend straightforward to see.

Here's the thought process:

• List the values of \(m\) for which you wish to find balances. In this case: 1, 3, 5, 7, 9, and 11.
• Plug each value of \(m\) back into the equation \(4800 - 400m\) to find the corresponding balance.
• Record these values into a structured table format to examine the trend.

For example:- At \(m=1\), the balance is \(4400\).- At \(m=7\), where the balance is precisely \(2000\), you can see how the equation predicts the trend over time.

This process not only provides specific balance figures but also forms the groundwork for graphing the function.

Functions

Functions are mathematical relationships where each input has a specific output. In our case, the function \(f(m) = 4800 - 400m\) shows how the balance in the account changes as months pass.

Each value of \(m\) (the number of months) gives exactly one balance (function output), depending on \(m\). This is an example of a linear function because it's a straight line when graphed.

Why are functions useful?

• They allow predictions about future balances, given any value of \(m\).
• Their properties, like slope and y-intercept, give us quick insights. Here, the slope is \(-400\), indicating a decrease in the balance by $400 each month.
• The y-intercept is 4800, which is the starting balance when \(m = 0\).

Understanding functions can help you predict and model real-world financial situations efficiently.