Unit 4 Systems of Equations

# Algebra Name: Date: Period: Unit 4 Systems of Equations (1) Page 401 – 402 #11 – 26 Left (2) Page 408 #14 – 29 Left (3) Page 408 #16 – 28 Right (4) Page 414 – 415 #8, 12, 19, 22, 25, 34 (5) Page 414 – 415 #11, 18, 24, 36, 39 ******Quiz Tomorrow****** (6) Page 429 – 430 #12 – 17 all, 18, 21 (7) Page 430 #19, 22, 23, 25, 26 (8) Page 409 #42, 43, Page 415 #43, 44 Page 422 #46 (9) Page 409 #44, Page 415 #45 – 47, Page 422 #47 (10) Page 409 #45, Page 415 #48, Page 422 #48 – 50 (11) Page 435 – 436 #9 – 14, 15 – 24 Left (12) Page 436 #16 – 25 Middle, 34 and 35 **(13) Problem Solving Worksheet *******Quiz Tomorrow******* (14) Test Review for Test Tomorrow (15) Page 444 – 445 #2 – 16 Even **Core Assessment Systems 1 7.1 Solving Linear Systems by Graphing (R,E/1) Two or more linear equations in the same variables is known as a __________________ of _________________ equations or simply a linear system. x + 2y = 5 2x – 3y = 3 Equation 1 Equation 2 A __________________ of this system of linear equations in two variables is an ordered pair (x,y) that satisfies both equations of the system. E1. Solve the linear system graphically. Check the solution algebraically. x + y = -2 2x – 3y = -9 SIF #1 SIF #2 m= b= m= Check: Equation 1 x + y = -2 Solution: ( b= Equation 2 2x – 3y = -9 , ) P1. Solve the linear system graphically. Check the solution algebraically. 6x – 3y = -15 2x + y = -3 2 7.2 Solving Systems by Substitution (R,E/2) Steps: 1. Pick an equation and a variable to isolate (1st or 2nd/ x or y) a. Choose the variable that can be isolated in one step or less b. Choose the variable that doesn’t create fractions when isolated 2. Substitute that expression into the opposite equation and solve 3. Use that value to solve for the missing variable 4. Check E1. Solve by Substitution: -x + y = 1 2x + y = -2 P1. Solve by substitution: 3x + y = 5 2x – y = 10 E2. Solve by substitution: 2x + 2y = 3 x - 4y = -1 P2. Solve by substitution: 2x + 6y = 15 x = 2y 3 7.3 Solving Systems by Linear Combinations (aka Elimination) (R,E/2) Steps: 1. Line up like terms in columns 2. Find or make opposites a. Multiply one or both equations by a number to obtain opposites 3. Add the equations together thus eliminating one of the variables and solve 4. Use that value to solve for the missing variable 5. Check E1. Solve by Elimination: 4x + 3y = 16 2x – 3y = 8 P1. Solve by Elimination: -x + 2y = -8 x + 6y = -16 E2. Solve by Elimination: 3x + 5y = 6 -4x + 2y = 5 P2. Solve by Elimination: 5x – 4y = 3 2x + 8y = -2 E3. Solve by Elimination: 3x + 2y = 8 2y = 12 – 5x P3. Solve by Elimination: 3x = - 6y + 12 -x + 3y = 6 4 7.5 Special Types of Linear Systems (I,E/2) Number of Solutions of a Linear System Lines Intersect One Solution-(x,y) Lines are Parallel No Solution Lines Coincide IMS-Any (x,y) on the line E1. Show that the linear system has no solution. 2x + y = 5 2x + y =1 Graphing Substitution E2. Show that the linear system has infinitely many solutions (IMS). Graphing Substitution 5 Elimination -2x + y = 3 -4x +2y = 6 Elimination P1. Solve the system and interpret the results: 3x + y = -1 -9x – 3y = 3 P2. Solve the system and interpret the results: x – 2y = 5 -2x + 4y = 2 P3. Solve the system and interpret the results: 2x + y = 4 4x – 2y = 0 6 7.4 and Supplement Applications of Linear Systems (I,E/3) Methods for Solving Systems: 1. Graphing-Good for providing a visual model or approximating a solution BUT it is not accurate. 2. Substitution-Good for systems that have a variable that is easy to isolate BUT can get complicated otherwise 3. Linear Combination (Elimination)-Accurate and efficient all of the time Example 1: Money Problem: A store sold 28 pairs of cross-trainer shoes for a total of $2220. Style A sold for $70 per pair and Style B sold for $90 per pair. How many of each style were sold? Chart: x: y: Verbal Model (System): System: Solve the System: Answer: 7 Example 2: Mixture problem: A coffee merchant has two types of coffee beans, one selling for $9 per pound and the other for $15 per pound. The beans are to be mixed to provide 100 pounds of a mixture selling for $13.50 per pound. How much of each type of coffee bean should be used to form 100 pounds of the mixture? Chart: x: y: Verbal Model (System): System: Solve the System: Answer: 8 Example 3: Making a Decision: You are offered two different sales jobs. Job A offers an annual salary of $30,000 plus a year-end bonus of 1% of your total sales. Job B offers an annual salary of $24,000 plus a year-end bonus of 2% of your total sales. a. How much would you have to sell to earn the same amount in each job? b. You believe you can sell between $500,000 and $800,000 of merchandise per year. Which job should you choose? Chart: x: y: Verbal Model (System): System: Solve the System: Answer Part (a): Answer Part (b): 9 Example 4: Distance/Rate Problems: A boat can travel 36 mi downstream in 2 hours. Coming back upstream the boat takes 3 hours. What is the rate of the boat in still water? What is the rate of the current? Chart: x: y: Verbal Model (System): System: Solve the System: 10 7.6 Systems of Linear Inequalities (Graphing) (I,E/2) Two or more linear inequalities in the same variables is known as a __________________ of _________________ inequalities or simply a system of inequalities. A _____________________ of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system. The ___________________ of a system of linear inequalities is the graph of all solutions of the system. E1. Graph: y<2 x ≥ -1 y>x-2 P1. Graph: x<4 y ≥ -3 x–y≥4 11 E2. Write a system of inequalities that defines the shaded region. P2. Write a system of inequalities that defines the shaded region. 12 E3. Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. x≥0 y≥0 y≤2 1 y ≤ − 2x + 3 P3. Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. x≥1 x≤4 y ≥ -1 1 y ≤ 2x + 1 13 E4. Lighting: You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium-throw spotlights and at least 1 long-throw spotlight. A medium-throw spotlight costs $1,000 and a long-throw spotlight costs $3500. The minimum order for free delivery is $10,000. a. Write and graph a system of linear inequalities that shows how many mediumthrow spotlights and long-throw spotlights should be ordered to get free delivery. b. Will an order of 4 medium-throw spotlights and 1 long-throw spotlight be delivered free? Chart: x: y: Verbal Model (System): [ ]≥[ ] [ ]≥[ ] [ ]∗[ ]+[ ]∗[ ]≥[ ] System: Answer (a) Graph: Answer (b): 14