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Math 803 Unit 7: Solving Systems of Equations 7.1 solve by graphing 7.2 solve by substitution 7.3 solve by elimination 7.4 special systems (0, 1, many solutions) 7.5 writing and solving systems (8.EE.7, 8.EE.8) (Textbook: Lessons 3.7-3.8) Name_____________________________ Period __________ Teacher __________________ 1 System of equations: two or more equations with the same set of variables. Example: y = 4x and y = 4x + 2 together are a system of equations. The solution for a system of equation is the point in which the two lines intersect. The solution is written as an ordered pair (x, y) For each problem determine if the ordered pair is the solution to the system of equations. Your answer will be YES or NO. SHOW ALL YOUR WORK to justify your answer. Example 1 Determine if the point (2, 7) is the solution to the following system of equations. y = 4x – 1 y = 2x + 3 YES…. (2,7) IS a solution for both lines. Example 2 Determine if the point (3, -1) is the solution of the system of equations? y=x–4 y = –3x Example 3 Determine if the point (-15, -25) is the solution of the system of equations? y = x – 10 y = 2x + 5 2 Practice: (substitute and solve) 1) Is the point (4, 12) a solution to the system 2) Is the point (2, 3) a solution to the system y = 3x 5x + 2y = 44 3) Is the point (3, 4) a solution to the system y = 2x + 7 3x – y = -9 4) Is the point (5, 2) a solution to the system x + 3y = 15 -8x + 3y = -12 5) Is the point (4, 6) a solution to the system 3x + y = 17 4x – y = 18 6) Is the point (3, 1) a solution to the system 4y + 4 = 7x 36 – 4y = 3x 7) Is the point (2, -1) a solution to the system 3x + 2y = 11 7x – y = 3 8) Is the point (5, -2) a solution to the system -4x – 9y = 1 -x + 2y = -4 x – 5y = 15 4x – 3y = 26 3 Lesson 7.1 - Solving Systems of Equations by graphing. a) b) c) d) Graph each equation on the same coordinate plane Find the point of intersection of the two lines Write this point as an ordered pair. This is the solution to the system Check your solution by substituting the coordinates into each equations Solve the system of equations by graphing each equation 1. 2. y=x–1 y = 2x - 2 3. y = 3x + 8 y=x–2 4 4. y = 2x – 4 1 y = 4 + 3 5. 3 y = −2 + 4 y= 3 2 −2 2 6. y = 3 + 4 y = -2x - 4 5 7. y = x − 4 y = −2x + 5 8. x – 4y = 4 5x – 4y = 12 9. y = -3x + 10 4y + 3x = 4 6 10. 5y = 2x – 10 y = -4x – 2 11. x – y = 4 x + 2y = -2 12. 4x – 5y = -28 -9x – 2y = 10 7 803 7.1a Skills Practice (8.EE.8) Solve Systems of Equations by Graphing Solve each system of equations by graphing. 1. y = x + 4 y = –2x – 2 2. y = 5x – 1 y = 5x + 10 3. y = x – 1 y – x = –1 4. y = 6x – 3 y = –3 5. CLUBS There are thirty-three students in the Chess Club. There are five more boys than girls in the club. Write and solve a system of equations to find the number of boys and girls in the Chess Club. 8 7.2 - Solving Systems of Equations by Substitution. 1. 2. 3. 4. 5. If needed, solve one of the equations for y (slope intercept form) Replace the y in one equation with the value of y for the other equation. Solve for x. Write answer as an ordered pair (x, y). This is the solution to the system Check your solution by substituting the coordinates into each equations Solve the system of equations by substitution. 1. 2. y = -3x + 1 y = -2 3. y = 7x – 20 y = -6 5. y = -6x y = -9x -15 7x + 2y = 11 y = -5 4. y = 9x y = 5x + 12 6. y = 4x + 25 y = 9x 9 7. 8. 3y = 9x 6x = 4y + 6 2x + y = -14 2y = 10x 7.2a Skills Practice (3.8) Solve each system of equations algebraically. 1. y = x – 8 y = 5x 2. y = –x – 4 y = 3x 3. y = x + 11 y = 12x 4. y = x – 14 y = –6x 5. y = –x + 9 y = 2x 6. y = x + 15 y = –4x 7. y = –x – 10 y = 4x 8. y = x + 24 y = –7x 9. y = –x + 18 y = 8x 10 7.2 - Solving Systems of Equations by Substitution. 1. 2. 3. 4. 5. 6. If needed, solve one of the equations for y (slope intercept form) Put parentheses around the expression you are substituting in to the other equation Replace this expression with the value of y for the other equation. Solve for x. Write answer as an ordered pair (x, y). This is the solution to the system Check your solution by substituting the coordinates into each equations 1. y = x + 12 4x + 2y = 12 2. 3. 4. y = x + 12 4x + 2y = 30 5. y = 2x – 3 x + y = 18 6. y = 2x + 1 -2x + 3y = -9 7. x + y = 1 5x – 4y = -14 8. -5x + y = 33 x – 2y = -21 y = 3x + 8 8x + 4y = 12 10 + 3y = 19 y = 2x + 5 11 803 Unit 7.1 – 7.2 Quiz REVIEW Solve each system of equations by graphing. 1. 2. y = 2 – 1 y=-x+2 y=x+4 y = –2x – 2 3. y = x – 1 y – x = –1 4. y = 6x – 3 y = –3 Use substitution to solve each system of equations. 5. y = 4x x+y=5 6. y = 2x x + 3y = –14 7. y = 3x 2x + y = 15 8. x = –4y 3x + 2y = 20 12 9. y = x – 1 x+y=3 10. x = y – 7 x + 8y = 2 11. 12. y = 3x + 8 5x + 2y = 5 y = 4x – 1 y = 2x – 5 13. Is (3,6) a solution to this system of equations? Show your work! 2x + 11y = 70 5x + 2y = 20 14. Is (7,4) a solution to this system of equations? Show your work! 4x + 9y = 64 6x + 2y = 50 15. Which of the following is a solution to this system of equations? Show your work! y = -3x + 6 y – x = -2 A. (0, 2) B. (-2, 0) C (2, 0) D. (0, -2) 13 7.3 - Solving Linear Systems by Elimination STEPS 1) Arrange the equations with like terms in columns x+y=5 x–y=7 x+y=5 x–y=7 2) Add the equations together. (This should SOLVE for one variable) 3) Substitute and solve for the other variable. 4) Check the solution in BOTH of the original equations. Example 1) 4x + 3y = 16 2x – 3y = 8 14 Example 2) x–y=8 x + y = 20 Example 3) x + 4y = 23 -x + y = 2 ***Example 4) 2y = x – 8 x + 6y = -16 Example 5) y=x-9 x + 8y = 0 15 7.3a Solve Systems by elimination Use elimination to solve each system of equations: 1) 2x + 2y = -2 3x – 2y = 12 2) 4x – 2y = -1 -4x + 4y = -2 4) -6x + 5y = 1 6x + 4y = -10 5) 2x – 3y = 12 4x + 3y = 24 7) x – y = 11 2x + y = 19 8) 8x = 9 – 5y -8x + 3y = 31 16 3) x – y = 2 x + y = -3 6) 2x + y = 9 3x – y = 16 9) 3x + 2y = 0 9x – 24 = 2y 7.3 - Solving by Elimination x + 3y = 7 3x + 3y = 9 (Day 2) What is different about these? Today we will work with linear equations where the coefficients are not the same. STEPS 1) Arrange the equations with like terms in columns **2) IF NEEDED - Multiply one or both of the equations by a number to get opposite numbers for one of the variables. 3) Add the equations together. (This should SOLVE for one variable) 4) Substitute and solve for the other variable. 5) Check the solution in BOTH of the original equations. 1) Solve the system x – y = -5 x + 2y = 4 17 2) Solve the system 5x + y = 9 10x – 7y = -18 3) Solve the system -7x + y = -19 -2x + 3y = -19 4) Solve the system 16x – 10y = 10 -8x – 6y = 6 5) Solve the system 6) Solve the system 8x + 14y = 4 -6x – 7y = 10 5x + 4y = 9 4x + 5y = 9 18 7) Solve the system 5x + 4y = -30 -3x – 9y = -18 8) Solve the system 3x = -6y + 12 x + 3y = 6 9) Solve the system 3x + 2y = 8 2y = 12 – 5x 10) Solve the system -2y = - 3x + 2 -5x – 5y = 10 19 7.3b PRACTICE Use elimination to solve each system of equations: 1) 3x + 2y = 0 x – 5y = 17 2) 2x + 3y = 6 x + 2y = 6 3) 3x – y = 2 x +2y = 3 4) 4x + 2y = 8 16x – y = 14 5) 4x + 5y = 6 6x – 7y = - 20 6) 10x + 3y = 19 y – 2x = 5 20 7.3c Exercises Use elimination to solve each system of equations. 1. 2x + 3y = 6 x + 2y = 5 2. 2m + 3n = 4 –m + 2n = 5 3. 3a – b = 2 a + 2b = 3 4. 4x + 5y = 6 6x – 7y = –20 5. 4x – 3y = 22 2x – y = 10 6. 3x – 4y = –4 x + 3y = –10 7. 4x – y = 9 5x + 2y = 8 8. 4a – 3b = –8 2a + 2b = 3 9. 2x + 2y = 5 4x – 4y = 10 10. 6x – 4y = –8 4x + 2y = –3 11. 4x + 2y = –5 –2x – 4y = 1 21 12. 2x + y = 3.5 –x + 2y = 2.5 7.3d Use elimination to solve each system of equations. 1. 2x – y = –1 2. 5x – 2y = –10 3. 7x + 4y = –4 3x – 2y = 1 3x + 6y = 66 5x + 8y = 28 4. 2x – 4y = –22 5. 3x + 2y = –9 6. 4x – 2y = 32 3x + 3y = 30 5x – 3y = 4 7. 3x + 4y = 27 5x – 3y = 16 10. 6x – 3y = 21 2x + 2y = 22 8. 0.5x + 0.5y = –2 x – 0.25y = 6 11. 3x + 2y = 11 2x + 6y = –2 22 –3x – 5y = –11 3 9. 2x – = –7 1 4 x+ =0 2 12. –3x + 2y = –15 2x – 4y = 26 Math 803 7.3-7.4 QUIZ REVIEW Use the elimination (linear combinations) method to solve the following linear systems: 1. x + y = 9 x–y=7 1._____________ 2. x – 2y = 8 -x + 3y = -5 2.______________ 3. 2x – 3y = -16 x + 3y = 10 3.________________ 23 4. -5x + 3y = 15 6x – 2y = -18 4.________________ 5. 4x – 5y = -18 5x + 4y = -2 5._______________ 6. 2x – y = 1 2x + 5y = -5 6._______________ 24 7.4 - Special Types of Linear Systems Possible Number of Solutions Two or more linear equations involving the same variables form a system of equations. A solution of the system of equations is an ordered pair of numbers that satisfies both equations. The table below summarizes information about systems of linear equations. Graph of a System Number of Solutions exactly one solution infinitely many solutions no solution Terminology consistent and independent consistent and dependent inconsistent GRAPHING Graph the following systems. Determine how many solutions the system has. If it has one solution give the coordinate. 1. y – 2x = 4 y = 2x 2. y = 3x – 4 1 y = −2 + 3 25 3. y = -2x + 4 y = -2x - 7 ALGEBRAICALLY (using substitution or elimination) When you get a TRUE statement : 3 = 3, or 0 = 0, or 12 = 12, etc… there are MANY solutions. When you get a FALSE statement: 0 = 4, 5 = 1, -3 = 10, etc… there is NO solution. Example 1 Pick the method of your choice. 2x + y = 5 2x + y = 1 Example 2: Pick the method of your choice. -x + 2y = -2 3x – 6y = 6 26 Example 3: Pick the method of your choice. -y = 3x + 4 -6x – 2y = -8 Example 4: Pick the method of your choice. 3x + y = -5 6x + 2y = 10 Example 5: Pick the method of your choice. 5x – y = 5 -x + 3y = 13 27 Practice: Solve by graphing 1. y = -3x + 4 -3y = 9x – 12 2. y = 5x – 2 -5y = -25x - 15 Solution ________________________ Solution ________________________ Solve Algebraically: Choose substitution or elimination 3. y = 3x + 4 y = 4x 4. -3x + 6y = 9 x + 2y = 5 28 5. y = 2x + 1 2y = 4x + 2 7.4a Solve each system by graphing Graph each system and determine the number of solutions it has. If it has one solution, name it. 1 1. y = –2 3x – y = –1 2. x = 2 2x + y = 1 3. y = x 2 x+y=3 4. 2x + y = 6 2x – y = –2 5. 3x + 2y = 6 3x + 2y = –4 6. 29 2y = –4x + 4 y = –2x + 2 7.4b Solve each system by graphing Graph each system and determine the number of solutions that it has. If it has one solution, name it. 1. 2x – y = 1 y = –3 2. x = 1 2x + y = 4 3. 3x + y = –3 3x + y = 3 4. y = x + 2 x – y = –2 5. x + 3y = –3 x – 3y = –3 6. y – x = –1 x+y=3 7. x – y = 3 8. x + 2y = 4 9. y = 2x + 3 x – 2y = 3 1 y=2x+2 30 3y = 6x – 6 7.4c Solve each system by graphing. 1. 3x – y = –2 3x – y = 0 4. x + y = 3 x + y = –3 2. y = 2x – 3 4x = 2y + 6 3. x + 2y = 3 3x – y = –5 5. 2x – y = –3 4x – 2y = –6 6. x + 3y = 3 x + y = –3 7. x + 3y = 3 2x – y = –3 31 7.5 – Writing and Using a Linear System in Real Life General Set-Up: x + y = __________ ___x + ___y = ________ $1590 was collected from 321 people at a museum. Adult tickets are $6 and child tickets are $4. How many adults and how many children went into the museum? 1) Define your variables. 2) Write your system by writing two different equations. 3) Solve: 4) Answer the question asked. 32 Example 1: An office supply company sells two types of fax machines. They charge $150 for one of the machines and $225 for the other. If the company sold 22 fax machines for a total of $3900 last month, how many of each type were sold? 1) Define your variables. 2) Write your system by writing two different equations. 3) Solve: 4) Answer the question asked. Example 2: Your math teacher tells you that next week’s test is worth 100 points and contains 38 problems. Each problem is either worth 5 points or 2 points. Because you are studying systems of linear equations, your teacher says that for extra credit you can figure out how many problems of each value are on the test. How many of each value are there? 33 3. Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 in operating costs per week plus $0.50 to produce each treat. He plans to sell each treat for $1.50. a. Graph the system of equations y = 0.5x + 20 and y = 1.5x to represent the situation. b. How many treats does Nick need to sell per week to break even? 4. A used book store also started selling used CDs and videos. In the first week, the store sold 40 used CDs and videos, at $4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $180.00 a. Write a system of equations to represent the situation. b. Graph the system of equations. c. How many CDs and videos did the store sell in the first week? 34 Example 3: Write and solve a system of equations that represents each situation. Sandy bought a total of 15 shirts and pairs of pants. She bought 7 more shirts than pants. How many of each did she buy? Example 4 Eight times a number plus five times another number is –13. The sum of the two numbers is 1. What are the numbers? Example 5 Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. Find the numbers. 35 7.5a Solving Real-Life Systems 1) Two times a number plus three times another number equals 13. The sum of the two numbers is 7. What are the numbers? 2) Four times a number minus twice another number is –16. The sum of the two numbers is –1. Find the numbers. 3) Two times a number plus three times another number equals 4. Three times the first number plus four times the other number is 7. Find the numbers. 4) The cost of 8 muffins and 2 quarts of milk is $18. The cost of 3 muffins and 1 quart of milk is $7.50. How much does 1 muffin and 1 quart of milk cost? Write an equation that represents the situation. Then solve a system of equation using the elimination method. 36 5) FUNDRAISING Trisha and Byron are washing and vacuuming cars to raise money for a class trip. Trisha raised $38 washing 5 cars and vacuuming 4 cars. Byron raised $28 by washing 4 cars and vacuuming 2 cars. Find the amount they charged to wash a car and vacuum a car. 6. Gregory’s Motorsports has motorcycles (two wheels) and ATVs (four wheels) in stock. The store has a total of 45 vehicles that, together, have 130 wheels. a. Write a system of equations that represents the situation. b. Solve the system of equations and interpret the solution. 7. The sum of two numbers is 28 and their difference is 4. What are the numbers? 37 7.5b Solving Real-Life Systems 1. The sum of two numbers is 41 and their difference is 5. What are the numbers? 2. Four times one number added to another number is 36. Three times the first number minus the other number is 20. Find the numbers. 3. One number added to three times another number is 24. Five times the first number added to three times the other number is 36. Find the numbers. 4. Find the two numbers whose sum is 29 and whose difference is 15. 5. The sum of two numbers is 24 and their difference is 2. What are the numbers? 38 6. Find the two numbers whose sum is 54 and whose difference is 4. 7. Two times a number added to another number is 25. Three times the first number minus the other number is 20. Find the numbers. 8. A roadside vegetable stand sells pumpkins for $5 each and squashes for $3 each. One day they sold 6 more squash than pumpkins, and their sales totaled $98. Write and solve a system of equations to find how many pumpkins and quash they sold? 39 9. INCOME Ramiro earns $20 per hour during the week and $30 per hour for overtime on the weekends. One week Ramiro earned a total of $650. He worked 5 times as many hours during the week as he did on the weekend. Write and solve a system of equations to determine how many hours of overtime Ramiro worked on the weekend. 10. BASKETBALL Anya makes 14 baskets during her game. Some of these baskets were worth 2-points and others were worth 3-points. In total, she scored 30 points. Write and solve a system of equations to find how 2-points baskets she made. 11. Creative Crafts gives scrapbooking lessons for $15 per hour plus a $10 supply charge. Scrapbooks Incorporated gives lessons for $20 per hour with no additional charges. Write and solve a system of equations that represents the situation. Interpret the solution. 40 803 Unit 7 - Systems Review Name__________________Period____ Solve each system by graphing. 1) y = -x + 2 y = 2x – 1 1) ________ 2) y = -2x + 2 x=3 2) ________ 3) y = -2x + 2 y=x+5 3) ________ 41 Solve each system by substitution. 4) y = -3x + 4 y = 3x - 2 4) ________ 5) y = -7 3x – 5y = 17 5) ________ 6) 3x – y = -14 y=x+4 6) ________ 42 Solve each system by linear combinations. 7) 7x + 2y = -19 x – 2y = -21 7) ________ 8) 6x – 12y = 24 2x + 12y =-8 8) ________ 9) 8x – 6y = -20 -16x + 7y = 30 9) ________ 10) 10) _______ 3x – 2y = 8 x – 2y = 0 43 Determine whether each of the following systems of equations have one solution, no solution or many solutions. 11) -4x - 4y = 10 2x + 2y = 20 11) ____________ 12) y = -4x + 6 y = -4x – 2 12) ____________ 13) x – 5y = 10 2x – 10y = 20 13) ____________ 14) y = 2x – 7 y = 2x – 7 14) ____________ 44 Set up and solve the following problems using systems of equations. 15) Jim runs a food cart and during a busy outdoor festival he sold $7343.75 worth of food. He sells hot dogs for $2.95 and steak sandwiches for $9.95. If he sold a total of 985 items that day, how many of each item did he sell? # of hot dogs: ________ # of steak sandwiches: ________ 16) Bob ate 10 carrots and 7 Hershey’s kisses and the total calories were 263. Fred ate 4 carrots and 8 Hershey’s kisses and the total calories were 230. How many calories are in a carrot and a Hershey’s kiss? (The answer is not a scientific truth, only made up numbers!!!) Calories in a carrot: _________ Calories in a Hershey Kiss: _________ 45