Problem 37 - Algebra 2

Question

Banking To pay your monthly bills, you can either open a checking account or use an online banking service. A local bank charges \(\$ 3\) per month and \(\$ .40\) per check, while an online services charges a flat fee of \(\$ 9\) per month. a. Write and graph a system of linear equations to model the cost \(c\) of each service for \(b\) bills that you need to pay monthly. b. Find the point of intersection of the two linear models. What does this answer represent? c. If you pay about 12 bills per month, which service should you choose? Explain.

Step-by-Step Solution

Answer

Graph two lines representing the costs: \(c = 3 + 0.40b\) and \(c = 9\). Their intersection point at \(b = 15\) bills indicates equal cost. For 12 bills, the local bank at \$7.80 is cheaper than the online service at \$9.

Step 1: Define the Variables

Let the variable 'c' represent the total cost, and let 'b' represent the number of bills paid each month.

Step 2: Set up the Equations

For the local bank, the total cost each month is given by the equation: \(c = 3 + 0.40b\). For the online banking service, the cost is a flat fee of \(c = 9\) per month, regardless of the number of bills.

Step 3: Graph the Equations

Plot both equations on the same graph to visualize the costs. The y-axis will represent the cost (c), and the x-axis will represent the number of bills (b). The local bank's equation will be a straight line starting at \(c = 3\) and with a slope of \(0.40\). For the online banking service, plot a horizontal line at \(c = 9\).

Step 4: Calculate the Intersection Point

To find the intersection of the two lines, set the two equations equal to each other: \(3 + 0.40b = 9\). Solve for 'b' to find the point of intersection.

Step 5: Solve for the Number of Bills

Subtract 3 from both sides of the equation: \(0.40b = 6\). Then, divide both sides by 0.40 to solve for 'b': \(b = \frac{6}{0.40} = 15\).

Step 6: Interpret the Intersection Point

The intersection point represents the number of bills where the cost of both services is the same, which is 15 bills.

Step 7: Determine the Better Service for 12 Bills

Calculate the total cost for 12 bills with both services: For the local bank, \(c = 3 + 0.40 \times 12 = 7.80\). For the online service, the cost remains \(c = 9\). Since \(7.80 < 9\), the local bank is cheaper.

Key Concepts

Graphing Linear Equations

The visualization of mathematical equations on a graph is a fundamental skill in algebra. When graphing linear equations, we essentially plot points on a plane that represent the solutions to those equations, creating a line that showcases all the possible solutions. For example, in the context of banking and monthly bills, we might consider a scenario where a bank charges a monthly fee plus an additional cost per transaction.

Take the equation for the bank's cost, which is given by
$$c = 3 + 0.40b$$.

Here, the cost c depends linearly on the number of bills b. To graph this equation, we start with the y-intercept at 3 (when b is zero). The slope of the line represents the rate of change of cost concerning the number of bills, which in this case, is 0.40. The line increases by 0.40 on the y-axis for every additional bill paid, showing a straight line with a positive slope on the graph.

A flat fee, like the one charged by the online banking service, will be represented by a horizontal line because the cost, c, remains constant regardless of the number of bills, which in our example is
$$c = 9$$

On the graph, a horizontal line crosses the y-axis at the cost of 9 and extends parallel to the x-axis, indicating that the output value never changes regardless of b.

Intersection Point of Lines

The point where two lines cross on a graph is known as the intersection point, and it can reveal significant information when comparing two linear models. In our banking example, finding the intersection point of the bank's fee structure and the online service's flat fee can help a customer understand at what point the costs are equivalent.

Graphically, locating the intersection point involves finding where the plotted lines of both services' costs meet on the graph. Algebraically, this point is found by equating the two equations. We solve
$$3 + 0.40b = 9$$

leading to the point of intersection where b represents the number of bills and c is the cost. When we solve the equation, we learn that the intersection occurs when 15 bills are paid monthly; at this point, both services cost the same amount. This intersection forms a coordinate (b, c) that can be identified on the graph.

Comparison of Linear Models

Comparing linear models help us understand which model or option is more beneficial under different circumstances. By plotting the equations on the same graph, as already done with the bank and online banking service, we can easily compare costs for different numbers of monthly bills paid.

Below are some steps highlighting how to compare linear models:

• Analyze the slope: A steeper slope indicates a faster increase in cost relative to the number of bills.
• Examine the intercept: Where the line crosses the y-axis tells us the initial amount paid or the fixed fee.
• Find the intersection: At what point do the two models cost the same? This is crucial for decision-making.
• Cheaper option: Check where each line lies for a specific number of bills to determine which service offers a lower cost.

When evaluating which service to choose for a specific number of bills (e.g., 12 monthly bills), calculate and compare the costs using each linear model. As we find out, a local bank would cost
$$c = 3 + 0.40 \times 12 = 7.80$$

while the online service remains at
$$c = 9$$.

Thus, for 12 bills, the local bank is the cheaper option. These visual and numerical comparisons provide a clear guide to making financially sound decisions based on one's individual needs.