Module 4: Linear Equa1ons Topic D: Systems of Linear Equa1ons and Their Solu1ons. Lesson 4-24: Introduc1on to Simultaneous Equa1ons

Transcription

1 Module 4: Linear Equa1ons Topic D: Systems of Linear Equa1ons and Their Solu1ons Lesson 4-24: Introduc1on to Simultaneous Equa1ons

2 Lesson 4-24: Introduc1on to Simultaneous Equa1ons Purpose for Learning: Students know that a system of linear equa5ons, also known as simultaneous equa5ons, is when two or more equa5ons are involved in the same problem and work must be completed on them simultaneously. Students also learn the nota5on for simultaneous equa5ons. Students compare the graphs that comprise a system of linear equa5ons, in the context of constant rates, to answer ques5ons about 5me and distance. Essen1al Ques1on(s): How can you iden5fy a solu5on to a system of linear equa5ons? Vocabulary: System of linear equa5ons, solu5on to a system of linear equa5ons, simultaneous equa5ons

3 Exercise 1: (in your student booklet) Derek scored 30 points in the basketball game he played and not once did he go to the free throw line. That means that Derek scored two point shots and three point shots. List as many combina5ons of two and three pointers as you can that would total 30 points. Two-Pointers Three-Pointers

4 Exercise 1: (in your student booklet) Derek scored 30 points in the basketball game he played and not once did he go to the free throw line. That means that Derek scored two point shots and three point shots. List as many combina5ons of two and three pointers as you 30 can that would total points. Three- Write an equa1on to describe the data. Two-Pointers Pointers Let x represent the number of 2 pointers and y represent the number of 3 pointers. 30=2x+3y

5 Exercise 2: (in your student booklet) Derek tells you that the number of two-point shots that he made is five more than the number of three-point shots. How many combina5ons can you come up with that fit this scenario? (Don t worry about the total number of points.) Two-Pointers Three-Pointers

6 Exercise 2: (in your student booklet) Derek tells you that the number of two-point shots that he made is five more than the number of three-point shots. How many combina5ons can you come up with that fit this scenario? (Don t worry about the total number of points.) Two- Pointers Three- Pointers Write an equa1on to describe the data. Let x represent the number of twopointers and y represent the number of three-pointers. x=5+y

7 Exercise 3: (in your student booklet) Which pair of numbers from your table in Exercise 2 would show Derek s actual score of 30 points? Two- Pointers Three- Pointers Two- Pointers Three- Pointers The pair 9 and 4 would show Derek s actual score of 30 points.

8 Simultaneous Linear Equa1ons: (Also known as a System of linear equa>ons.) ---Situa1ons where working with two linear equa1ons simultaneously is necessary. The situa5on with Derek can be represented as a system of linear equa5ons: Let x represent the number of two-pointers and y represent the number of three-pointers, then { 2x+3y=30 x=5+y The nota5on for simultaneous linear equa5ons let s us know that we are looking for the ordered pair (x, y) that makes both equa5ons true. That point is called the solu1on to the system. Just like equa5ons in one variable, some systems of equa5ons have exactly one solu1on, no solu1on, or infinitely many solu1ons. Ul5mately our goal is to determine the exact loca1on on the coordinate plane where the graphs of the two linear equa1ons intersect, giving us the ordered pair (x, y) that is the solu1on to the system of equa1ons.

9 Simultaneous Linear Equa1ons: (Also known as a System of linear equa>ons. We can graph both equa5ons on the same coordinate plane: The situa5on with Derek can be represented as a system of linear equa5ons: Let x represent the number of two-pointers and y represent the number of threepointers, then { 2x+3y=30 x=5+y Note the point of intersec5on. Does it sa5sfy both equa5ons in the system? The point of intersec>on of the two lines is (9, 4). 2(9)+3(4)=30 9= =30 9=9 30=30 Yes, x=9 and y=4 sa>sfies both equa>ons of the system. Derek made 9 two-point shots and 4 three-point shots. It is the solu1on to the system of linear equa1ons. It is the one point that makes both equa1on in the system true.

10 Exercise 4: (in your student booklet) Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels 45 miles per hour for the en5re trip. Fernie is 70 not such a safe driver. He drives miles per hour throughout the trip. Fernie and Efrain leo from the same loca5on, but Efrain leo at 8:00 a.m. and Fernie leo at 11:00 a.m. Assuming they take the same route, will Fernie ever catch up to Efrain? a.) Write If so, the approximately linear equa1on when? that represents Efrain s constant speed. Make sure to include in your equa1on the extra 1me that Efrain was able to travel. Efrain s average speed: y/x = miles/hour so C = 45/1 miles per hour, which is the same as 45 miles per hour. If y represents the distance he travels in x hours, then we have y/x =45 and the linear equa>on y=45x. To account for his addi>onal 3 hours of driving >me that Efrain gets, we write the equa>on y=45(x+3) y=45x+135 (Efrain s equa1on)

11 Exercise 4: (con1nued) Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels 45 miles per hour for the en5re trip. Fernie is not such a safe driver. He drives 70 miles per hour throughout the trip. Fernie and Efrain leo from the same loca5on, but Efrain leo at 8:00 a.m. and Fernie leo at 11:00 a.m. Assuming they take the same route, will Fernie ever catch up to Efrain? b.) Write If so, the approximately linear equa1on when? that represents Fernie s constant speed. Fernie s average speed over one hour is 70/1 miles per hour, which is the same as 70 miles per hour. If y represents the distance he travels in x hours, then we have y/x =70 and the linear equa>on y=70x. y=70x (Fernie s equa1on) c.) Write the system of linear equa1ons that represents this situa1on. { y=45x+135 y=70x

12 Exercise 4: (con1nued) Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels 45 miles per hour for the en5re trip. Fernie is 70 not such a safe driver. He drives miles per hour throughout the trip. Fernie and Efrain leo from the same loca5on, but Efrain leo at 8:00 a.m. and Fernie leo at 11:00 a.m. Assuming they take the same route, will Fernie ever catch up to Efrain? d.) Sketch If so, the approximately graph. when? { y=45x +135 y=70x

13 Exercise 4: (con1nued) Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels 45 miles per hour for the en5re trip. Fernie is 70 not such a safe driver. He drives miles per hour throughout the trip. Fernie and Efrain leo from the same loca5on, but Efrain leo at 8:00 a.m. and Fernie leo at 11:00 a.m. Assuming they take the same route, will Fernie ever catch up to Efrain? e.) Will If Fernie so, approximately ever catch when? up to Efrain? If so, approximately when? Yes, Fernie will catch up to Efrain aier about 4 1/2 hours of driving or aier traveling about 325 miles. f.) At approximately what point do the graphs of the lines intersect? The lines intersect at approximately (4.5, 325). Therefore this point is the solu>on to this system as it is the only point that makes both equa>ons in the system true. { y=45x +135 y=70x

14 Exercise 5: (in your student booklet) Jessica and Karl run at constant speeds. Jessica can run 3 miles in 15 minutes. Karl can run 2 miles in 8 minutes. They decide to race each other. As soon as the race begins, Karl realizes that he did not 5e his shoes properly and a.) Write the linear equa1on that represents Jessica s constant speed. 1 takes Make sure minute to include to fix them. in your equa1on the extra 1me that Jessica was able to run. Jessica s average speed over 15 minutes is 3/15 miles per minute, which is equivalent to 1/5 miles per minute. If y represents the distance she runs in x minutes, then we have y/x = 1/5 and the linear equa1on y= 1/5 x. To account for her addi1onal 1 minute of running that Jessica gets, we write the equa1on y= 1/5 (x+1) y= 1/5 x+ 1/5 (Jessica s equa1on)

15 Exercise 5: (con1nued) Jessica and Karl run at constant speeds. Jessica can run 3 miles in 15 minutes. Karl can run 2 miles in 8 minutes. They decide to race each other. As soon as the race begins, Karl realizes that he did not 5e his shoes properly and b.) Write the linear equa1on that represents Karl s constant speed. 1 takes Karl s minute average to fix speed them. over 8 minutes is 2/8 miles per minute, which is the same as 1/4 miles per minute. If y represents the distance he runs in x minutes, then we have y/x = 1/4 and the linear equa1on y= 1/4 x. y= 1/4 x (Kar l s equation) c.) Write the system of linear equa1ons that represents this situa1on. { y= 1/5 x+ 1/5 y= 1/4 x

16 Exercise 5: (con1nued) 3 15 Jessica and Karl run at constant speeds. Jessica can run miles in 2 8 minutes. Karl can run miles in minutes. They decide to race each other. As soon as the race begins, Karl realizes that he did not 5e his shoes properly and d.) Sketch the graph. 1 takes minute to fix them. { y= 1/5 x+ 1/5 y= 1/4 x

17 Exercise 5: (con1nued) e.) Use the graph to answer the ques1ons below. 1. If Jessica and Karl raced for 2 miles. Who would win? Explain. If the race were 2 miles, then Karl would win. It only takes Karl 8 minutes to run 2 miles, but it takes Jessica 9 minutes to run the distance of 2 miles. 2. If the winner of the race was the person who got to a distance of 1/2 mile first, who would the winner be? Explain. At 1/2 miles, Jessica would be the winner. She would reach the distance of 1/2 miles between 1 and 2 minutes, but Karl wouldn t get there un>l about 2 minutes have passed. 3. At approximately what point would Jessica and Karl be 1ed? Explain. Jessica and Karl would be >ed aier about 4 minutes or a distance of 1 mile. That is where the graphs of the lines intersect.

18 Lesson Summary: Simultaneous linear equa5ons, or a system of linear equa5ons, is when two or more linear equa5ons are involved in the same problem. Simultaneous linear equa5ons are graphed on the same coordinate plane. The solu5on to a system of linear equa5ons is the set of all points that make the equa5ons of the system true. If given two equa5ons in the system, the solu5on(s) must make both equa5ons true. Systems of linear equa5ons are iden5fied by the nota5on used, for example: { y= 1/8 x+ 5/2 y= 4/25 x