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.

Outlines

00:00

📈 Linear Depreciation Modeling

This paragraph introduces the concept of linear modeling through word problems involving linear equations. The example provided is about a fax machine purchased for $2,700 that depreciates to a value of $100 after 5 years. The goal is to determine its value after 3 years. The explanation covers identifying points on the graph, calculating the slope using the formula (y1 - y2) / (x1 - x2), and then using the slope-intercept form of the equation (y = mx + b) to find the value of y when x equals 3. The result shows that the fax machine was worth $1,140 after 3 years. The paragraph also briefly mentions another example involving business sales growth over years, encouraging viewers to apply the same method to solve similar problems.

05:05

📊 Predicting Future Sales with Linear Equations

The second paragraph continues the theme of linear modeling, focusing on predicting future sales based on past data. It uses the example of a business that had sales of $50,000 in the second year and $100,000 in the fifth year, aiming to predict sales in the eighth year. The process involves calculating the slope of the sales growth, using the point-slope form of the equation (y - y1 = m(x - x1)) to find the equation of the sales line. The calculation simplifies to find the predicted sales for the eighth year, resulting in an expected sales figure of $150,000. The paragraph concludes with an invitation for viewers to try a similar problem on their own, using a car depreciation example as a prompt, and emphasizes the importance of finding the correct points, slope, and equation to solve the problem.

Keywords

💡Linear Modeling

Linear modeling is a mathematical method used to understand relationships between variables by fitting a linear equation to data points. In the video, linear modeling is applied to word problems involving depreciation of assets and sales growth, helping to predict future values based on historical data.

💡Word Problems

Word problems are practical scenarios presented in narrative form that require the application of mathematical concepts to find a solution. The video script uses word problems to demonstrate how linear equations can be used to solve real-world issues, such as determining the value of a fax machine over time.

💡Linear Equations

Linear equations are algebraic expressions that represent a straight line when graphed, typically in the form of y = mx + b, where m is the slope and b is the y-intercept. The video explains how to set up and solve linear equations to find unknown variables in various contexts.

💡Depreciation

Depreciation refers to the decrease in the value of an asset over time due to wear and tear or obsolescence. The video script discusses the depreciation of a fax machine, using linear modeling to calculate its worth after a certain number of years.

💡Slope

In the context of linear equations, the slope (m) represents the rate of change between the dependent and independent variables. The video demonstrates calculating the slope as the difference in y-values divided by the difference in x-values, which is essential for determining the equation of the line.

💡Y-Intercept

The y-intercept (b) is the point where the line crosses the y-axis on a graph. It represents the value of y when x is zero. In the video, the y-intercept is used as a starting point to establish the linear equation for the value of the fax machine.

💡Sales Growth

Sales growth refers to the increase in the volume of sales over a period. The script uses linear modeling to predict future sales figures based on past sales data, illustrating how businesses can use this information for planning and forecasting.

💡Point-Slope Form

The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. The video explains how to use the point-slope form when the y-intercept is not known, as in the case of predicting future sales.

💡Asset Value

Asset value is the worth of an item owned by a company or individual. The video script discusses calculating the depreciated value of a fax machine as an asset, showing how its value changes over time.

💡Business Forecasting

Business forecasting involves predicting future trends and conditions based on current data and knowledge. The video uses linear modeling as a tool for forecasting sales figures in a business context, demonstrating its practical application for strategic planning.

Highlights

Introduction to linear modeling with word problems using linear equations.

Problem setup: A fax machine bought for $2,700 and its depreciation over time.

Identifying the initial value of the fax machine at 0 years as $2,700.

Understanding the independent variable (time) and dependent variable (money) in the context of depreciation.

Establishing the final value of the fax machine after 5 years as $100.

Calculating the slope of the depreciation using the two points: (0, 2700) and (5, 100).

Determining the slope as 520, indicating the rate of depreciation per year.

Formulating the linear equation using the slope and y-intercept.

Using the linear equation to find the value of the fax machine after 3 years.

Calculating the value of the fax machine after 3 years to be $1,140.

Transition to a second example involving business sales over years.

Analyzing sales growth from the second to the fifth year of business.

Finding the slope of sales growth using the points (2, 50000) and (5, 100000).

Using point-slope form to establish the linear equation for sales.

Predicting sales in the eighth year of business using the linear equation.

Calculating the predicted sales in the eighth year to be $150,000.

Encouraging the audience to try a similar problem involving the depreciation of a car.

Providing guidance on how to approach the car depreciation problem.

Emphasizing the importance of finding the slope and equation to solve for the unknown variable.

Transcripts

hello today we're going to be working on

linear modeling so that's word problems

with linear equations our first question

a company buys a fax machine for

$2,700 if it is worth 100 at the end of

5 years what was the machine worth after

3 years so we want to find our points

first so we have they bought a fax

machine at

2,700 so when it was 0 years

old it was worth 2,700

so our independent variable X is time

and our dependent variable Y is money

because how much it's worth depends on

how old it is now if it's worth 100 at

the end of five years so our New point5

Years

$100 what will the machine be worth

after or what was the machine worth

after 3 years so when it's three years

how much will it be so because we have

two points we can find our

slope and we know that slope is y1 - Y 2

over X1 - X2 so we just fill this in

using our first two points so we get

2700 -

100

over 0 - 5 which gives us 2600

/5 which is equal

to5 20 so that is our slope remember m

equals

slope so now we can write our equation

because we have our Y intercept here so

remember that's our B so we have y

equals our slope

M

X+ B which is

2700 in this

case and then we need to find out when X

is 3 what will y be so y =

520 X is 3 +

2700 So then we multiply y =

1560 +

2700 we simpli y this and we get

11,040 so our answer when X is 3 y will

be

1,140 that's our answer and remember

it's in

dollars because it is an amount so what

was the machine worth after 3 years the

machine was worth

$1,140 let's look at another example

if a business had sales of 50,000 the

second year of business and 100,000 in

sales the fifth year how much will their

sales be in the eighth year of business

so again we're going to look at those

points so they had a sale of 50,000 in

the second year so remember the sales

depends on the year so we have second

year was

50,000 and 100 ,000 the fifth year so

five years they do

100,000 how much will their sales be in

the eighth year of business so in eight

years how much will they make again we

can find our

slope by using the first two points so

we do our y

50,000 minus 100,000

/ 2 - 5 which is

50,000 / -3 negatives cancel so our

slope is

50,000 divided by

3 now we don't have our Y intercept this

time so we're going to have to use point

slope form so we say y minus we'll use

this first

equation for the point so y -

50,000 equals our slope

50,000 /

3 time x -

2 remember we're using those Point

values and we have to find out

when X is 8 what will y be so y -

50,000 =

50,000 over 3 * 8 - 2 8 - 2 is 6 so y -

50,000 = 50,000 / 3 we multiplying by

six now the three and the six can

simplify or reduce that becomes a two

so y -

50,000 = 50,000 * 2 which is

[Music]

100,000 and then we just have to add our

50,000 so we get y =

150,000 okay and that is our answer for

the eth year of

business so what I'd like you to do is

try one on your own so a car was

purchased for

$188,000 after 5 years it was worth

$10,500 when will it be worth nothing so

remember you need to find your points

first so look at the information and get

your two points then try to find out

what you're trying to find are you

looking for the X are you looking for

the Y what information do you have find

the equation to use okay make sure you

find your slope then find the equation

and then you can solve for what you're

looking for good luck