Problem 66 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. The value of a building bought in 1995 may be depreciated (or decreased) as time passes for income tax purposes. Seven years after the building was bought, this value was \(\$ 225,000\) and 12 years after it was bought, this value was \(\$ 195,000\). a. If the relationship between number of years past 1995 and the depreciated value of the building is linear, write an equation describing this relationship. Use ordered pairs of the form (years past \(1995,\) value of building). b. Use this equation to estimate the depreciated value of the building in 2013 .

Step-by-Step Solution

Answer

The equation is \( y = -6,000x + 267,000 \). In 2013, the building's value is estimated to be \$159,000.

Step 1: Define ordered pairs

Identify the ordered pairs (years past 1995, value of building) from the problem. We have (7, 225,000) and (12, 195,000).

Step 2: Determine the slope

The slope of a line is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points (7, 225,000) and (12, 195,000), find the slope: \( m = \frac{195,000 - 225,000}{12 - 7} = \frac{-30,000}{5} = -6,000 \).

Step 3: Use a point to find the y-intercept

Use the slope \( m = -6,000 \) and one of the points, say (7, 225,000), to find the y-intercept \( b \) using the equation \( y = mx + b \). Plugging in the values: \( 225,000 = -6,000(7) + b \). Solve for \( b \): \( 225,000 = -42,000 + b \) which gives \( b = 267,000 \).

Step 4: Write the equation in slope-intercept form

Now that you have both the slope \( m = -6,000 \) and y-intercept \( b = 267,000 \), write the equation of the line in slope-intercept form: \( y = -6,000x + 267,000 \).

Step 5: Estimate the value in 2013

2013 is 18 years past 1995. Use the equation \( y = -6,000x + 267,000 \) and plug in 18 for \( x \) to estimate the building's value: \( y = -6,000(18) + 267,000 = -108,000 + 267,000 = 159,000 \).

Key Concepts

Linear Relationship

A linear relationship exists when there is a consistent rate of change between two variables. This means that as one variable changes, the other variable changes at a constant rate. In the context of this exercise, the linear relationship describes how the value of the building decreases over time.
Understanding linear relationships is crucial in many fields because it simplifies the prediction of future events based on past data. In mathematical terms, a linear relationship between two variables can be represented by a straight line on a graph. This line can be described by the equation of a line in the slope-intercept form. This form is given by the equation:

• \( y = mx + b \)

Here, \( m \) represents the slope and \( b \) represents the y-intercept.

Depreciation

Depreciation refers to the loss of value of an asset over time. This is particularly common with physical assets like buildings or vehicles, where consistent use and time decrease their worth. It's a concept that's not just reserved for theory—it plays a significant role in accounting and tax calculations.
In exercise terms, the building is losing value as years pass by, starting from the year it was bought in 1995. By 2002, the building's value was \\(225,000, and by 2007, it had depreciated to \\)195,000. This decrease in value over the years is what we call depreciation.
Depreciation can often be expressed as a linear relationship, especially in simplified models. This means identifying a consistent rate at which the value decreases, which corresponds directly to the slope in the slope-intercept form of a line.

Equation Writing

Writing equations helps to succinctly express relationships between variables in a form that's easy to understand and manipulate. For linear relationships, writing the equation involves determining the slope and the y-intercept. These are the two core components we need.
From the exercise, we first identify two points that the line passes through. These are (7, 225,000) and (12, 195,000). Using these, we calculate the slope \( m \) as follows: \\[ m = \frac{195,000 - 225,000}{12 - 7} = -6,000 \]Using this slope, we use one ordered pair to find the y-intercept by applying it in the equation form:

• \( y = mx + b \)

Thus, substituting the knowns to find \( b \), we end up with the equation \( y = -6,000x + 267,000 \). This gives a complete expression of how the building's value decreases annually.

Ordered Pairs

Ordered pairs in math are used to represent data points on a graph. They are essentially coordinate points formatted as (x, y). These are useful in understanding relationships in two-dimensional space. In this exercise, ordered pairs like (7, 225,000) and (12, 195,000) represent actual, measurable data.
In the problem, the first number in each pair indicates years past 1995, while the second number shows the corresponding value of the building at that time. These points help us to plot a straight line to represent how the value progresses with time.

• Ordered pairs are essential because they are easy to visualize and allow one to see patterns and trends.
• They also help in performing calculations, such as determining the slope, which is a core component in writing linear equations.

Knowing how to recognize and use ordered pairs is key in solving linear problems efficiently.