Linear Modeling

Summary

This video script delves into linear modeling through word problems involving linear equations. It begins with a scenario of a fax machine's depreciation, illustrating how to determine its value after 3 years using two points and calculating the slope. The script then transitions to predicting business sales, demonstrating the use of the point-slope form of a linear equation. Finally, it challenges viewers to apply these concepts to a car depreciation problem, encouraging them to find the equation and solve for when the car will be worth nothing.

Takeaways

• 📈 The video discusses solving word problems using linear equations, specifically for depreciation and sales growth over time.
• 💰 The first example involves a fax machine bought for $2,700 that depreciates to $100 in 5 years, and the task is to find its value after 3 years.
• 📊 The script introduces the concept of independent variable (X for time) and dependent variable (Y for value) in the context of linear equations.
• 🔍 To find the value of the fax machine after 3 years, the script uses the slope formula, which is the change in Y over the change in X.
• 📐 The slope is calculated using the initial and final points, which are (0, $2,700) and (5, $100), resulting in a slope of $520 per year.
• 📉 The linear equation is then formed using the slope and the Y-intercept, which in this case is the initial value of the machine.
• 🧮 The script demonstrates solving for Y when X is 3 years, resulting in a value of $1,140 for the fax machine at that time.
• 📈 The second example deals with the sales growth of a business, starting at $50,000 in the second year and increasing to $100,000 by the fifth year.
• 📊 The script again uses the slope formula to determine the rate of sales increase, which is $50,000 per year.
• 🔢 The point-slope form of the equation is used to find the sales in the eighth year, which is calculated to be $150,000.
• 🚗 The final task given to the viewer is to apply the same method to determine when a car purchased for $188,000 will be worth nothing, based on its depreciation to $10,500 after 5 years.
• 📝 The script emphasizes the importance of identifying the correct points, understanding the variables, calculating the slope, and using the appropriate equation to solve for the unknown.

Q & A

What is the initial cost of the fax machine the company bought?

The company initially bought the fax machine for $2,700.

What is the value of the fax machine at the end of 5 years?

At the end of 5 years, the fax machine is worth $100.

What is the slope of the depreciation line for the fax machine?

The slope of the depreciation line is calculated as (2700 - 100) / (0 - 5) = $520 per year.

What is the equation of the depreciation line for the fax machine?

The equation of the depreciation line is y = 520x + 2700, where y is the value of the fax machine and x is the time in years.

What was the value of the fax machine after 3 years?

After 3 years, the fax machine was worth 520 * 3 + 2700 = $1,140.

What is the sales amount of the business in the second year?

In the second year of business, the sales amount was $50,000.

What is the sales amount of the business in the fifth year?

In the fifth year of business, the sales amount was $100,000.

What is the slope of the sales growth line for the business?

The slope of the sales growth line is calculated as (100000 - 50000) / (5 - 2) = $50000 per year.

How can you find the sales amount in the eighth year of business using the point-slope form?

Using the point-slope form, y - 50000 = (50000 / 3) * (x - 2), and solving for y when x is 8 gives y = 150000.

What is the method to find when the car purchased for $188,000 will be worth nothing?

To find when the car will be worth nothing, first determine the two points of the depreciation line, calculate the slope, use the slope to find the equation, and then solve for the time when the value (y) is zero.

What is the independent variable in the linear modeling of the fax machine's depreciation?

The independent variable is time (x), as the value of the fax machine depends on how old it is.

What is the dependent variable in the linear modeling of the fax machine's depreciation?

The dependent variable is the money (y), representing the value of the fax machine at a given time.