Writing and Solving equations with years

Writing and Solving equations with years

Lesson Objective: 4.01a Use linear functions or inequalities to model and solve problems; justify results

Students will know how to use base year analysis to write and solve word problems with years

Writing and Solving equations with years

When comparing information from two years, we can use base-year analysis

We make the first year of information year zero

To get the second year we subtract the second year from the first year

Writing and Solving equations with years

Year 1: (1995, 25) Year 2: (2010, 55)

Base year: year 1 = 0 so we have (0, 25)

Year 2 = 2010 – 1995 = 15, so: (15, 55)

We can than use the two points to get a slope and an equation

Writing and Solving equations with years

(0, 25), (15,55)

Slope equation: m =

Replace y2 with 55 and y1 with 25 Replace x2 with 15 and x1 with 0 Simplify the top and the bottom Reduce the fraction We have a slope of 2

y2 – y1

x2 – x1

55 - 2515 – 0 30152

Writing and Solving equations with years

Once you know the slope, plug it into the equation

Replace the y and x with one of the points.

I would suggest the first point since x = 0

Multiply and solve for b Plug b back into the equation

y = mx + by = 2x + b25 = 2(0) + b25 = 0 + b25 = by = 2x + 25

Writing and Solving equations with years

Eric Cartman High School had 1500 students in 2000 and 1600 in 2005. Assuming a linear increase, how many students will be in Cartman High in 2011?

“Respect my authoriti and learn!”

Writing and Solving equations with years

Eric Cartman High School had 1500 students in 2000 and 1600 in 2005. Assuming a how many students will be in Cartman High in 2011?

“linear increase” means what? Slope!

linear increase,

Writing and Solving equations with years

Slope equation: m =

What do we need to find the slope? Two sets of ordered pairs

Writing and Solving equations with years

Eric Cartman High School had 1500 students in 2000 and 1600 in 2005. Assuming a linear increase, how many students will be in Cartman High in 2011?

Years will always be x, so replace x1 with 2000 and x2 with 2005

(x, y), (x, y)x1 , y1 x2 , y2(2000, y), (2005, y)

Writing and Solving equations with years

Eric Cartman High School had 1500 students in 2000 and 1600 in 2005. Assuming a linear increase, how many students will be in Cartman High in 2011?

When comparing years we can call the first year zero (0) and the next year 5 in this case

x1 , y1 x2 , y2(2000, y), (2005, y)(0, y), (5, y)

Writing and Solving equations with years

Eric Cartman High School had 1500 students in 2000 and 1600 in 2005. Assuming a linear increase, how many students will be in Cartman High in 2011?

Replace the y’s with the value goes with each year

x1 , y1 x2 , y2(0, y), (5, y)(0, 1500), (5, y)(0, 1500), (5, 1600)

Writing and Solving equations with years

Replace the y’s in the equation with the numbers in the ordered pairs

(0, 1500), (5, 1600)

y2 – y1

x2 – x1m =

1600 - 1500

Writing and Solving equations with years

Replace the x’s in the equation with the numbers in the ordered pairs

(0, 1500), (5, 1600)

x2 – x1m =

1600 - 15005 - 0

Writing and Solving equations with years

Simplify the top and the bottom Simplify the fraction. If it’s not an

even number, leave it as a fraction in it’s lowest form

m =1600 - 1500

5 - 0100

520

Writing and Solving equations with years

Once you know the slope, plug it into your equation:

Next we must find b. To find b we plug in either point for

x and y

y = mx + by = 20x + b

Writing and Solving equations with years

Plug in the first point Multiply on the right side 20(0) cancels out so we’re left with

1500 = b

y = 20x + b

(0, 1500), (5, 1600)

1500 = 20x + b1500 = 20(0) + b1500 = 0 + b1500 = b

Writing and Solving equations with years

Plug b into the equation

The question then asks, “how many students will be in Cartman High in 2011?”

X is always years, so we’ll plug in the year for x. Remember, we have to plug in how many years it’s been since 2000, so plug in 11

y = 20x + 1500y = 20x + by = 20(11) + 1500

Writing and Solving equations with years

Multiply 20(11) Add together to get your answer In 2011 there should be 1720

students at Cartman High

y = 20(11) + 1500y = 220 + 1500y = 1720

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

Linear equation means y = mx + b First we need to find the m

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been each year. Write a linear equation to represent how

are in Kenny City for any year.

Years are always x, so what goes with years in the problem?

many people

decreasing by 2,000

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

Decreasing means subtracting, so replace m with -2,000

y = mx + by = -2000x + b

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

Next we need to find the b We plug in a point to solve for b

y = -2000x + b

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

When we deal with years, the first year is called year zero, so 1990 is year 0

y = -2000x + b

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

Replace y with 170,000 and x with 0

y = -2000x + b170000 = -2000(0) + b

Writing and Solving equations with years

Multiply -2000(0) 170000 = b Plug b back into the equation

170000 = -2000(0) + b170000 = 0 + b170000 = by = -2000x + 170000

Writing and Solving equations with years

In 1990 there were 170,000 people in Kenny City. Since then, the population has been decreasing by 2,000 each year. Write a linear equation to represent how many people are in Kenny City for any year.

y = -2000x + 170000

Writing and Solving equations with years

If this trend continues, how many people will be left in Kenny City in 2015?

Use the equation to plug in the new year

Remember, 1990 was year 0, so 2015 would be year 25

y = -2000x + 170000y = -2000(25) + 170000

Writing and Solving equations with years

Multiply -2000(25) Add the numbers together y = 120000, so in 2015 there will be

120,000 people left in Kenny City

y = -2000(25) + 170000y = -50000+ 170000y = 120000

Writing and Solving equations with years

Stan Marsh Ski Slope had 6000 skiers for the season in 1980. In 2000 it had 9600 skiers. If Stan Marsh Ski Slope continues to increase at the same rate, how many skiers will there be in 2011?

Writing and Solving equations with years

Broflovski’s law firm handled 524 cases in 1995. In 2003 they handled 628 cases. Assuming a linear increase, how many cases should they have handled in 2010?

Writing and Solving equations with years

In 1980, the average apartment at Butters Apartments was $250. By 2004, the average price was $702. (Let x = 80 represent 1980) Create a linear model that best represents this situation then find the average price of an apartment in 2010.