Problem 32 - Introductory and Intermediate Algebra for College Students 4th

Question

In \(2008,\) rent payments averaged 824 dollar per month. For the period shown, monthly rent payments increased by approximately 7 dollar per year. If this trend continues, how many years after 2008 will rent payments average 929 dollar In which year will this occur?

Step-by-Step Solution

Answer

15 years after 2008, i.e., in 2023, the average monthly rent will be $929.

Step 1: Identify the given information and the quantities to find

The given information in the exercise is that in 2008, the rent payment was $824 per month and it increased by approximately $7 per year. You are asked to find out how many years after 2008 will the rent payments average $929 and in which year will this occur.

Step 2: Formulate the linear equation

Let's define 'x' as the number of years since 2008. Therefore, the rent 'r' in any given year would be calculated as r = 824 + 7x. This equation represents the relationship between the years since 2008 and the rent, showing an increase of $7 per year.

Step 3: Solve the equation

We need to find when the rent 'r' will be $929. Therefore, we will set r to 929 and solve for x. So, 929 = 824 + 7x. Subtract 824 from both sides to get: 105 = 7x. Then, divide both sides by 7 to isolate x: x = 15.

Step 4: Interpret the result

The solution x = 15 means that 15 years after 2008, the monthly rent will average $929. To find the year, simply add 15 to 2008, which gives 2023.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, used to express relationships between variables through straight-line graphs. They are represented in the standard form of \( y = mx + b \) where:

• \( y \) is the dependent variable.
• \( m \) is the slope or rate of change.
• \( x \) is the independent variable.
• \( b \) is the y-intercept or starting point.

The problem you've encountered here is a typical application of a linear equation. It shows how rent increases every year by a fixed amount. When we identify the starting rent in 2008 as 824 dollars and acknowledge the steady 7 dollar yearly increase, we formulate the equation \( r = 824 + 7x \). Here:

• \( r \) is the rent after \( x \) years.
• \( x \) represents the years since 2008.

Linear equations like this one are straightforward to solve once the values for the variables are plugged in, allowing us to project future scenarios, like in this case.

Problem Solving

Problem-solving is all about breaking down a problem into manageable parts. In this exercise, we first identify known quantities, like the baseline rent and annual increase, and our target rent, which is 929 dollars.
The subsequent step involves formulating a plan by converting these knowns into a mathematical equation. This involves expressing the trend in rent as a linear equation, which we can then manipulate to find the number of years after the initial year (2008).
To solve for \( x \), the number of years, we begin with the equation \( 929 = 824 + 7x \). Subtracting 824 from 929 isolates the term involving \( x \), leading to 105 = 7x. Dividing by 7 gives \( x = 15 \).
This step-by-step approach helps untangle the complexity, bringing clarity and logic to problem-solving through mathematics.

Mathematical Modeling

Mathematical modeling is a fascinating yet practical method used to represent real-world situations through mathematical concepts. This approach can predict future outcomes or solve complex problems. In your task, modeling the rent increase involves translating everyday scenarios, like rent changes, into mathematical expressions.
The equation \( r = 824 + 7x \), created for this context, is a basic model showing the trend in rent payments. It captures the pattern of rent increasing by 7 dollars every year.
Through solving this model, we predict that the rent will reach 929 dollars 15 years after 2008. By adding 15 to 2008, we determine that this takes place in 2023.
Mathematical models like these allow us to interpret data clearly and make informed decisions or projections about the future.