Writing Linear Equations from a Context Algebra 1

Learning Targets Write a linear equation in slope-intercept form from a context containing 2 ordered pairs. Write a linear equation in slope-intercept form from a context containing a slope and a point. Interpret the components of the slope-intercept equation in a context.

Guidelines Points: describes an instant with 2 “units” Example: After 3 hours, he earned $30. 3, 30 Slope: describes a behavior over time with 2 “units”. Key word: “per” Example: She makes $17 per hour. 𝑚= $17 1 ℎ𝑜𝑢𝑟 = 17 1 =17

Example 1 Greg is driving a remote control car at a constant speed. He starts the time when the car is 5 feet away. After 2 seconds the car is 35 feet away. A) Write an equation in slope intercept form to represent the situation where 𝑑 represents the distance and 𝑡 represents time. Point 1: (0, 5), Point 2: (2, 35) 𝑚= 35−5 2−0 = 30 2 =15 𝑑−5=15 𝑡−0 𝑑−5=15𝑡 𝑑=15𝑡+5

Example 1 Continued Greg is driving a remote control car at a constant speed. He starts the time when the car is 5 feet away. After 2 seconds the car is 35 feet away. B) Estimate the distance the car has traveled after 10 seconds. 𝑑=15𝑡+5 𝑑=15 10 +5 𝑑=150+5=155 𝑓𝑒𝑒𝑡

Example 2 During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July. A) Write a linear equation, in slope intercept form, to predict Malik’s Cost of gasoline, the first of any month during the year, using 1 to represent January, where 𝐶 is the cost and 𝑡 is the month. Point 1: (6, $3.20), Point 2: (7, $3.42) 𝑚= 3.42−3.20 7−6 = .22 1 =.22 𝐶−3.20=.22(𝑡−6) 𝐶−3.20=.22𝑡−1.32 𝐶=.22𝑡+1.88

Example 2 Continued During one year, Malik’s cost for self-serve regular gasoline was $3.20 on the first of June and $3.42 on the first of July. B) What was the cost of gasoline on the first of December? 𝐶=.22𝑡+1.88 𝐶=.22 12 +1.88 𝐶=$4.52

Example 3 Ana is driving from her home in Miami, Florida, to her grandmother’s house in New York City. On the first day, she will travel 240 miles to Orlando, Florida, to pick up her cousin. Then they will travel 350 miles each day. A) Write an equation, in slope intercept form, that models the total number of miles 𝑚 Ana has traveled, if 𝑑 represents the number of days after she picks up her cousin. Point: (1, 240), Slope: 350𝑚𝑖𝑙𝑒𝑠 1𝑑𝑎𝑦 =350 𝑚−240=350 𝑑−1 𝑚−240=350𝑑−350 𝑚=350𝑑−110

Example 3 Continued Ana is driving from her home in Miami, Florida, to her grandmother’s house in New York City. On the first day, she will travel 240 miles to Orlando, Florida, to pick up her cousin. Then they will travel 350 miles each day. B) How far does she travel after 4 days? 𝑚=350𝑑−110 𝑚=350 4 −110 𝑚=1290 miles

Example 4 In 1991, 1267 manatees inhabited Florida’s waters. The manatee population has decreased at a rate of 123 manatees per year. A) Write an equation for the manatee population (𝑃), 𝑡 years after 1991. Point: 0, 1267 , Slope: − 123 𝑚𝑎𝑛𝑎𝑡𝑒𝑒𝑠 1 𝑦𝑒𝑎𝑟 =−123 𝑃−1267=−123 𝑡−0 𝑃−1267=−123𝑡 𝑃=−123𝑡+1267

Example 4 Continued In 1991, 1267 manatees inhabited Florida’s waters. The manatee population has decreased at a rate of 123 manatees per year. B) What year will the population be 898 manatees? 𝑃=−123𝑡+1267 898=−123𝑡+1267 −369=−123𝑡 𝑡=3 years after 1991 Answer: year 1994

Example 5 The equation 𝐶=7𝑡+20 can be used to determine 𝐶, the total cost in dollars, of a babysitter that works 𝑡 hours. What does the 7 and 20 represent in this context? Slope: 7 7 1 = $7 1hour Y-Intercept: (0, 20) (0 ℎ𝑜𝑢𝑟𝑠, $20) The babysitter earns $20 upfront and then, an additional $7 per every hour worked.

Example 6 The equation 𝑇=3ℎ+4 represents the total amount of water, in inches, inside of a pool after ℎ hours of rain. Describe what 3 and 4 represent in this context. Slope: 3 1 = 3 𝑖𝑛𝑐ℎ𝑒𝑠 1 ℎ𝑜𝑢𝑟 Y-intercept: (0 ℎ𝑜𝑢𝑟𝑠, 4 𝑖𝑛𝑐ℎ𝑒𝑠) There are 4 inches of water inside the pool before it starts raining. After it starts raining, water collects inside the pool at 3 inches per hour.

Example 7 The table below shows the total amount of a snow accumulating on a driveway over time. The data can be modeled by a linear equation where 𝑥 is the elapsed minutes and 𝑦 is the total amount of snow on the driveway in centimeters. A) What does the y-intercept of the linear equation that models the data indicate? Y-Intercept: 0, 2 →(0 minutes, 2 cm) There were 2 centimeters of snow on the driveway when it starting snowing. 𝒙 𝒚 1 5 2 8 3 11