7.1 Solving Systems of Equations: Graphing Name Part I - Warm Up with ONE EQUATION: a. Which of the following is a solution to the equation: y 3x 1? a. (-2, -3) b. (1, 3) c. (2, 5) d. (-2, -6) Partt II TWO EQUATIONS a. Can you find a solution that works in BOTH of the givenn equations? 10x 2 y 8 y x 2 a. (-2, 6) b. (-1, 1) c. (1, 3) d. (0, 0) b. Graph the equation. Confirm that the point you chose above lies on the line. Label the point on your graph. y b. Graph both equations (plot at least 4 points per line). y x x c. Choose another possible solution from your graph. Plug in this point to verify algebraically thatt it is in fact a solution. c. Where do you find your solution from part a? d. Are there any other points that are solutions to BOTH equations? Why or Why Not? d. How many possible solutions are there? in a full sentence. Explain why Check with teacher Check with teacher 1
Part III: Application #1 You signed up for an internet movie company called WebFlix. The website has a flat monthly fee of $6, plus a charge of $3 for each new release. a. Write an equation that models the situation. Define your variables! Partt IV: Application #2 Briann and Kelly bring their nephews and nieces to a carnival. Brian buys 4 bags of cotton candy and 2 burgers and spends $20. Kelly buys 6 bags of cotton candy and 9 burgers and spends $54 dollars. How muchh was each bag of cotton candy and each burger? a. Write two equations that model this situation. b. Graph the equation. b. Graph both equations. y x c. What does the point (2,12) represent? d. In general what does each point on the line represent? c. What does the solution represent? e. What does the point (-3, -3) represent? d. How many solutions are there? Why? f. How many solutions are there? Why? Check with teacher Check with teacher 2
Partner Practice Solve the linear system by graphing. Check your solution. 2x + y = 9 x - 3y = 6 2x + 3y = 15 2x - 3y = 3 -x + y = -2 2y + 4x = 12 2x y = 6 2x y = -10 3
7.2 Solving Systemss of Equations: Substitution Collision Road Rage Activity Car A begins at position 0 and drives to the right. Car B begins at position 100 and drives to the left. Answer the following questions to find where and when the cars will collide. 1. Assume that car A travels 5 units per second and begins att position 0. Where will the car be after 10 seconds? 2. a. Write an equation in slope intercept form that cann be used to calculate the position of car A after x seconds. b. Explain the meanings of x, y, slope, and y interceptt in terms of the problem situation. 3. Assume that car B travels 4 units per second and begins att position 100. Where will the car be after 10 seconds? 4. a. Write an equation in slope intercept form that cann be used to calculate the position of car B after x seconds. b. Explain the meanings of x, y, slope, and y interceptt in terms of the problem situation. 5. Calculate when and where the cars will crash into each other using the equations you found. 4
Bandana Fundraiser Activity Aidan has an idea that could raise money for the freshman class. He would like to sell bandanas for the winter pep rally to show that the freshman class has the most school spirit! 1. Plot points representing supply for each price in the table. Draw the line through the data points, and write Supply on this line. Selling Price of Each Button $1.000 $2.000 $4.000 Number of Buttons in Stock Supply 35 130 320 Number of Buttons that Students will Buy 530 400 140 2. Plot points representing the number of bandanas requested demand for each selling price on the same graph. Draw the line through these points. Label this line Demand. 3. If Aidan sets the price at $2.50 per bandana, how many disappointed customers can he expect to have? Explain how you got your answer. 4. If Aidan sets the price at $3.80 per bandana, how many unsold bandanas can he expect to have left over? Explain how you got your answer. 5. If Aidan gives the buttons away at no charge, how many bandanas would he need? How does the graph help you determinee your answer? 6. What price would make the bandana supply so low that the number of available bandanas would be zero? 7. Estimate the price at whichh supply and demand will be in equilibrium. Whatt is this price and how many bandanas can Aidan expect to sell? How does the graph help you determine your answer? 8. Use your graph to find the equation for supply S as a function of price P. 9. Use your graph to find the equation for demand D as a function of price P. 10. Solve the system of supply and demand equations to find the price and the number of bandanas that Aidan should order for supply and demand to be in exact equilibrium. How does thiss price compare with your answer in Question 7? 5
7.2 Day 2 Solving Systems of Equations: Substitution Partner Activity! 1) Solve the system of Equations by Graphing: 2) Solve the system of Equations by Graphing: y x7 x 4y 8 2x 3y 9 y x7 Think and Discuss: Did you experience any frustrations with the problems above? Describe in a couple sentences below: Alternatives to Graphing: SUBSTITUTION METHOD! Think and Discuss: What does the word SUBSTITUTE mean? Solve the following using the Substitution Method. y x7 x 4y 8 2x 3y 9 y x7 6
Does this appear to be the correct point on the graph above? Steps for SUBSTITUTION METHOD! 1.. Decide whichh variable is easiest to. 2.. for that variable. 3.. the expression from step 2 back into the OTHER equation. 4. Use the found value of one variable to find the value of the other variable. 5. both values by plugging them back into the original equation. Let s Practice Substitution Ex. 1 x2y 7 x y 23 Ex. 2 7x 2y 24 4x y 8 Ex. 3 y 2x 19 y x 7 Ex. 4 y 2x y 3x 2 Ex. 6 Gina went shopping for holiday presents. She bought boxes of chocolates and boxes of ornaments for her coworkers. Boxes of chocolate cost $8 each and boxes of ornaments costt $6 each. She buys a total of 22 boxes and spends $152. How many boxes of chocolates and how many boxes of ornaments does she buy? 7
7.3 Solving Systems of Equations: Elimination Example 1: The sum of two numbers is -5, and the difference of the two numbers is -17. What are the two numbers? a) Set up an equation for the sum of two numbers. b) Set up an equation for the difference of the two numbers. c) Can you figure out how to solve to find the two numbers? Example 2: Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold? a) Set up an equation for the two small pitchers and one large pitcher. b) Set up an equation for the one large pitcher and one small pitcher. c) Use the two equations to find how many cups of water each pitcher can hold. 8
Example 3: Bill and Steve decide to spend the afternoon at an amusement park enjoying their favorite activities, the waterr slide and the gigantic Ferris wheel. Their tickets are stamped each time they slide or ride. At the end of the afternoon they have the following tickets: a) Based on what we did for the last two problems, set up two equations (one for Bill and one for Steve) to solvee the system of linear equations by elimination. b) How much does it cost to slide on the Water Slide? How much does it cost on the ferris wheel? Example 4: On a typical day with light winds, the 1800 mile flight from Charlotte, North Carolina, to Phoenix, Arizona, takes longer than the return trip because the plane has to fly into the wind. (Distance = rate x time) a) The flight from Charlotte to Phoenix is 4 hours 30 minutes long, and the flight from Phoenix to Charlotte is 4 hours long. Find the average speed (in miles per hour) of the airplane on the way to Phoenix and on the return tripp to Charlotte. b) Let s be the speed (in miles per hour) of the plane with no wind, and let w be the speed (in miles per hour) of the wind. Use your answer to part (a) to write and solve a system of equationss to find the speed of the plane with no wind and the speed of the wind. 9
Example 4: Solve the linear system using elimination. a. x 5y 9 4x 5y 14 b. 3x 4y 2 3x 2y 26 You try! c. 8x 3y 12 8x 9y 12 d. 5x 6y 4 7x 6y 8 Example 5: Solve the linear system using elimination (Arrange like terms) a. 6x + 7y = 16 b. 4x 5y = 5 y = 6x 32 5y = x + 10 10
7.4 Solving Systems of Equations: Elimination Example 1: The play, Noises Off, costs $5 for students/seniors and $10 for adults. 500 tickets were sold for a total of $4,085. How many student/senior tickets were sold? How many adult tickets were sold? Example 2: Jacob and Cody decide to go to Taco Bell for lunch. Jacob ordered 3 soft tacos and 3 burritos for $11.25. Cody ordered 4 soft tacos and 2 burritos for $10. How much does each soft taco cost? How much does each burrito cost? Example 3: The Detroit Pistons are on a winning streak! Crazy, right? In one of their last games, they scored a total of 127 points in the game and 52 total baskets. If they made 11 free throws, how many 2 point and 3 point shots were made? Example 4: Solve the linear system using elimination (multiply one equation, then add or subtract) a. 3x 3y = 21 b. 2x + y = -9 8x + 6y = -14 4x + 11y = 9 11
Example 5: Solve the linear system using elimination (Multiply both equations, then add or subtract) a. 7x + 2y = 26 b. 3y = -2x + 17 10x 5y = -10 3x + 5y = 27 Partner Practice! Solve the linear system using elimination 1. 6x 2y = 1 2. 2x + 5y = 3-2x + 3y = -5 3x + 10y = -3 3. 3x 7y = 5 4. 2x 3y = 6 9y = 5x + 5 4y = -7x 8 12
7.5 Solving Special Types of Linear Systems Warm Up with your Partner: One admission to an ice skating rink costs x dollars and renting a pair of skates costs y dollars. A group pays $243 for admission for 36 people and 21 skate rentals. Another group pays $81 for admission for 12 people and 7 skate rentals. Determine the cost of admission and the cost of renting skates. (1-2) Warm Up: Graph the following systems of equations, then record your observations. 1. 2 y x4 3 2 y x2 3 a. Where do your lines intersect? b. How can you be sure? c. What do you think the solution to the system is? Why? 2. 6x 8y 40 3x 4y 20 a. Where do your lines intersect? b. How can you be sure? d. What do you think the solution to the system is? Why? 13
3. Discuss your findings above with your partner: then sketch a graph of a system with two lines that has: a. one solution b. no solutions c. infinitely many solutions Recognizing Special Cases Algebraically y Part I: a. Graph ANY two lines that are parallel. b. Write the equation of these two lines in slope intercept form. x c. Solve the system of equations using substitution. d. What do you discover? Explain why are you ending up with this solution. Part II: a. Solve the following system using elimination: y x + 3y = 15-2x 6y = -30 x b. Based on the result above, what do you predict is the solution? c. What would you predict this looks like graphically? d. Graph the system of equations. 14
Part III: Practice Solve the following systems algebraically and find the solution. Then state what the system looks like graphically. a. 3x + 2y = 10 b. y = 7x + 4-3x - 2y = - 2-21x + 3y = 12 Part IV: Now What? Consider the system: 2 y x 4 3 3x y 5 y a. Can you find the solution to the system by graphing? Why or Why not? x b. Solve the system algebraically: c. Did you experience any frustrations solving algebraically? Could you have found that answer graphically? Part V: Journal Questions: 1. Write a summary describing what you learned today about linear systems and their solutions (both graphically and algebraically). 15
Section 6.7: Graph Linear Inequalities in Two Variables Linear inequality in two variables: replace the = sign in a linear equation with <., >, or. Example 1: Tell whether the ordered pair is a solution of the inequality a. 3x 4y > 9 (2, 0) b. 2x + 3y 14 (5, 2) c. y 8 (-9, -7) 16
Steps for graphing a linear inequality in two variables: Step 1: Put the inequality into a nice graphing formant, and then graph the boundary line. Use a line for < or > Use a line for or Step 2: Determine which side of the line to shade, and then shade that entire region. Shade the line for < or Shade the line for > or Example 2: Graph the inequality a. y < - 1 2 x + 4 b. y 3x + 1 c. 2y + 4x > 8 d. x + 4y < - 8 177
e. y < 2 f. x 1 g. y > -2x + 3 h. x 4 18
Section 7.6: Solve Linear Systems of Linear Inequalities Systems of linear inequalities Solutions of a system of linear inequalities Graph of a system of linear inequalities Example 1: Graph the system of inequalities y > -x 2 Example 2: y < 3x y 3x + 6 y -2x + 1 199
Example 3: x + y 5 Example 4: y > 1 y < x + 3 x 4 3y < 6x 6 Try : Graph the system of inequalities y < x 4 x > -2 y -x + 3 y 4 3x + 4y 24 20