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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
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WWhen comparing information from two years, we can use base-year analysis WWe make the first year of information year zero TTo get the second year we subtract the second year from the first year
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YYear 1: (1995, 25) YYear 2: (2010, 55) BBase year: year 1 = 0 so we have (0, 25) YYear 2 = 2010 – 1995 = 15, so: (15, 55) WWe can than use the two points to get a slope and an equation
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((0, 25), (15,55) SSlope equation: m = Replace y 2 with 55 and y 1 with 25 RReplace x 2 with 15 and x 1 with 0 SSimplify the top and the bottom RReduce the fraction WWe have a slope of 2
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OOnce you know the slope, plug it into the equation RReplace the y and x with one of the points. II would suggest the first point since x = 0 MMultiply and solve for b PPlug b back into the equation
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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!”
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EEric 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? SSlope! linear increase,
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SSlope equation: m = WWhat do we need to find the slope? TTwo sets of ordered pairs
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EEric 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? YYears will always be x, so replace x 1 with 2000 and x 2 with 2005 x 1, y 1 x 2, y 2
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EEric 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? WWhen comparing years we can call the first year zero (0) and the next year 5 in this case
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EEric 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? RReplace the y’s with the value goes with each year
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RReplace the y’s in the equation with the numbers in the ordered pairs
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RReplace the x’s in the equation with the numbers in the ordered pairs
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SSimplify the top and the bottom SSimplify the fraction. If it’s not an even number, leave it as a fraction in it’s lowest form
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OOnce you know the slope, plug it into your equation: NNext we must find b. TTo find b we plug in either point for x and y
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PPlug in the first point MMultiply on the right side 220(0) cancels out so we’re left with 1500 = b
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PPlug b into the equation TThe question then asks, “how many students will be in Cartman High in 2011?” XX 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
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MMultiply 20(11) AAdd together to get your answer IIn 2011 there should be 1720 students at Cartman High
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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.
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IIn 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. LLinear equation means y = mx + b FFirst we need to find the m
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IIn 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. YYears are always x, so what goes with years in the problem? many people decreasing by 2,000
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IIn 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. DDecreasing means subtracting, so replace m with -2,000
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IIn 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. NNext we need to find the b WWe plug in a point to solve for b
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IIn 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. WWhen we deal with years, the first year is called year zero, so 1990 is year 0
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IIn 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. RReplace y with 170,000 and x with 0
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MMultiply -2000(0) 1170000 = b PPlug b back into the equation
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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.
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IIf this trend continues, how many people will be left in Kenny City in 2015? UUse the equation to plug in the new year RRemember, 1990 was year 0, so 2015 would be year 25
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MMultiply -2000(25) AAdd the numbers together yy = 120000, so in 2015 there will be 120,000 people left in Kenny City
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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?
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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?
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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.
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