Problem 61 - College Algebra

Question

Last year, at Haven's Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of \(\$ 140,000\) . This year, due to inflation, the same cars would cost \(\$ 151,830\) . The cost of the BMW increased by \(8 \%,\) the Jeep by 5\(\%\) and the Toyota by 12\(\% .\) If the price of last year's Jeep was \(\$ 7,000\) less than the price of last year's BMW, what was the price of each of the three cars last year?

Step-by-Step Solution

Answer

BMW: $49,090.91, Jeep: $42,090.91, Toyota: $48,818.18.

Step 1: Define Variables

Let the price of the BMW last year be \( x \), the price of the Jeep last year be \( y \), and the price of the Toyota last year be \( z \). We are given two pieces of information that help us form equations: \( y = x - 7000 \) and \( x + y + z = 140,000 \).

Step 2: Adjust Current Prices

Using the percentage increases, the price of each car this year can be written as follows: the BMW is \( 1.08x \), the Jeep is \( 1.05y \), and the Toyota is \( 1.12z \). Together this totals \( 151,830 \), giving us the equation \( 1.08x + 1.05y + 1.12z = 151,830 \).

Step 3: Substitute Known Values

Substitute \( y = x - 7000 \) into both equations. This yields two equations: 1) \( x + (x - 7000) + z = 140,000 \) and 2) \( 1.08x + 1.05(x - 7000) + 1.12z = 151,830 \).

Step 4: Simplify Equations

Simplify the first equation: \( 2x + z = 147,000 \). Simplify the second equation: \( 1.08x + 1.05x - 7350 + 1.12z = 151,830 \) which simplifies further to \( 2.13x + 1.12z = 159,180 \).

Step 5: Solve for 'z' in terms of 'x'

Using the first simplified equation, express \( z \) in terms of \( x \): \( z = 147,000 - 2x \).

Step 6: Substitute 'z' into Second Equation

Plug \( z = 147,000 - 2x \) into the second equation: \( 2.13x + 1.12(147,000 - 2x) = 159,180 \). Simplify this to get: \( 2.13x + 164,640 - 2.24x = 159,180 \).

Step 7: Solve for 'x'

Simplify to find \( x \): \( -0.11x + 164,640 = 159,180 \). Solving for \( x \) gives \( x = 49,090.91 \).

Step 8: Calculate 'y' and 'z'

Using \( y = x - 7000 \) gives \( y = 42,090.91 \). Substitute \( x \) back into \( z = 147,000 - 2x \) to get \( z = 48,818.18 \).

Step 9: Verify the Solution

Check that \( x + y + z = 140,000 \) holds true: \( 49,090.91 + 42,090.91 + 48,818.18 = 140,000 \). Verify the adjusted totals match \( 151,830 \): \( 1.08 \times 49,090.91 + 1.05 \times 42,090.91 + 1.12 \times 48,818.18 = 151,830 \), confirming the correct prices.

Key Concepts

Inflation Impact on Prices

Inflation refers to the general increase in prices over time, which decreases the purchasing power of money. When inflation occurs, the same amount of money buys fewer goods and services. For businesses like the Haven's Pond Car Dealership, this can mean that the prices of their cars increase to keep up with the general rise in the cost of goods. In the context of the exercise, inflation has caused the BMW, Jeep, and Toyota models to increase their prices by different percentages. This reflects the typical impact of inflation on consumer prices, where each type of item may have a unique rate of price increase. It is important to monitor inflation rates because they directly affect both consumers and companies, like car dealerships, in adjusting their pricing strategies.

System of Linear Equations

A system of linear equations consists of two or more equations that have the same set of unknowns. In this exercise, the prices of the three cars last year form such a system. We are given two equations:

• Equation 1: \( x + y + z = 140,000 \) - representing the total price of all three cars last year.
• Equation 2: \( 1.08x + 1.05y + 1.12z = 151,830 \) - representing the total price this year after inflation.

Additionally, we have a relationship between the variables: \( y = x - 7000 \), which helps us reduce the number of unknowns. By using these equations together, we can solve for the individual prices of the BMW, Jeep, and Toyota last year. Such systems are common in algebra and allow us to find relationships between different variables to solve real-world problems.

Algebraic Substitution Method

The algebraic substitution method is a technique used to solve systems of equations. It involves substituting one equation into another, effectively reducing the system to fewer unknowns. In our exercise, we use the relationship \( y = x - 7000 \) to substitute for \( y \) in both the original equations. This simplification step replaces two of the unknowns with one in each equation, making it easier to solve. By expressing \( y \) in terms of \( x \), and later finding \( z \), the system becomes easier to handle, and directly solving for one variable becomes possible. Once one variable is found, the others can be calculated easily, completing the solution. This approach is widely used in algebra because it leverages known relationships and simplifies complex systems efficiently.

Percentage Increase Calculations

Percentage increase calculations are crucial for determining how much a value has changed over time. In this exercise, calculating the increased prices of the cars involves applying the respective inflation rates to last year's prices:

• The BMW's price increases by \(8\%\),
• The Jeep's by \(5\%\), and
• The Toyota's by \(12\%\).

To calculate a percentage increase, we multiply the original price by the percentage increase expressed as a decimal. For example, the new price of a BMW is calculated as \( 1.08x \). Similarly, for the Jeep, it is \( 1.05y \), and for the Toyota, \( 1.12z \). These calculations are important in many fields, such as finance and economics, where understanding growth or inflation is key to making informed decisions.