Read To Go On? Skills Intervention 3-1 Using Graphs and Tables to Solve Linear Sstems Find these vocabular words in Lesson 3-1 and the Multilingual Glossar. Vocabular sstem of equations linear sstem consistent sstem inconsistent sstem independent sstem dependent sstem Solving Linear Sstems b Using Graphs and Tables Use a graph and a table to solve 3 2 6. Solve each equation for. Subtract the -term from both sides to isolate. 3 2 6 2 2 5 3 6 Plot each line on the grid. Complete the table of values for each equation. 3 or 2 6 or 6 0 3 0 6 1 2 3 4 1 2 3 4 5 What point do the lines have in common? (, ) The solution to the sstem is (, ). Classifing Linear Sstems Classif the sstem 3 2 4, and determine the number of solutions. 6 4 8 Solve each equation for. 3 2 4 6 4 8 2 4 6 4 3 3 2 Do the equations have the same slope? The same -intercept? Are the sstems dependent? How man solutions are there? Copright b Holt, Rinehart and Winston. 43 Holt Algebra 2
Read To Go On? Skills Intervention 3-2 Using Algebraic Methods to Solve Linear Sstems Find these vocabular words in Lesson 3-2 and the Multilingual Glossar. Vocabular substitution elimination Solving Linear Sstems Using Substitution Use substitution to solve the sstem of equations 2 2 1. STEP 1 The first equation is alread solved for which variable? What does equal in the first equation? STEP 2 Substitute the first equation from Step 1 into the second equation for. STEP 3 Solve for. 2 1 2 1 STEP 4 Solve for the other variable. 2 1 2 1 2 2 2 The solution to the sstem of equations is,. Solving Linear Sstems b Elimination Use elimination to solve the sstem of equations 2 10. 6 STEP 1 To eliminate, multipl the first 1( 2 ) 10( 1) equation b. STEP 2 Combine the two equations using. STEP 3 Solve for. STEP 4 Solve for the other variable. 6 6 6 6 6 4 The solution to the sstem of equations is,. Copright b Holt, Rinehart and Winston. 44 Holt Algebra 2
Read To Go On? Skills Intervention 3-3 Solving Sstems of Linear Inequalities Find this vocabular word in Lesson 3-3 and the Multilingual Glossar. Vocabular sstem of linear inequalities Graphing Sstems of Inequalities Graph each sstem of inequalities. A. 2 3 3 Should the boundar line for 2 be solid or dashed? Draw the boundar line 2 on the graph. Should ou shade above or below the boundar line? 5 Shade the region on the graph. Should the boundar line for 3 3 be solid or dashed? Draw the boundar line 3 3 on the graph. Should ou shade above or below the boundar line? Shade this region on the graph. What part of the graph shows the solution? Check the point (0, 5). Does this make the sstem true? 2 4 B. 3 1 Draw the boundar line for 2 4 on the graph. Shade the region the boundar line. Draw the boundar line for 3 on the graph. Shade the region to the Draw the boundar line for 1 on the graph. Shade the region of the boundar line. the boundar line. Test the point (0, 2) from the overlapping region to check the solution. 2 4 2 2(0) 4; 2 3 0 1 2 1 5 Copright b Holt, Rinehart and Winston. 45 Holt Algebra 2
A sstem of linear inequalities is two or more inequalities graphed on the same coordinate sstem. The solution is represented b the overlapping region. As a fundraiser, the swim team sells tacos and nachos. The make $1 for ever taco and $2 for ever nacho the sell. The club cannot sell more than 100 tacos or 150 nachos. The club s goal is to make at least $200 in total profit. Write and graph a sstem of inequalities that models this situation. Understand the Problem 1. How much mone does the team earn from each taco? From each nacho? 2. How man tacos do the have available to sell? Nachos? 3. How man inequalities need to be written? Make a Plan Let represent the number of tacos and represent the number of nachos the sell. 4. Complete: Read To Go On? Problem Solving Intervention 3-3 Solving Sstems of Linear Inequalities The team cannot sell more than 100 tacos. The team cannot sell more than 200 nachos. Profit from tacos Profit from nachos 200 1 5. Plan how to graph each inequalit. Inequalit Boundar Line Dashed or Solid Line Shaded Region vertical line at line Shade to the of the line. Horizontal line at line Shade the line. 1 -intercept: (200, 0) -intercept: (0, 100) line Shade above the line. Solve 6. Graph each inequalit and shade the correct regions. 7. Are the following points located in the intersection of all three regions? (50, 100), (75, 150), and (80, 175) 250 Look Back 8. Are each of the -coordinates less than 100? 9. Are each of the -coordinates less than 200? 10. Does each point make the inequalit 2 200 true? Copright b Holt, Rinehart and Winston. 46 Holt Algebra 2 250
Read To Go On? Skills Intervention 3-4 Linear Programming Find these vocabular words in Lesson 3-4 and the Multilingual Glossar. Vocabular linear programming constraint feasible region objective function Solving Linear Sstems b Using Graphs and Tables Graph each feasible region, and maimize or minimize the objective function P 3 4. 1 A. minimize; 0 1 Graph each inequalit on the grid. Shade the feasible region in which the three inequalities overlap. What are the three vertices of the feasible region? ( 1, ), (, ), and (, ) Tr each point in the objective function. 5 5 Verte ( 1, 0) P 3 4 P 3( 1) 4(0) 3 P 3(1) 4(2) 11 Which verte results in the lowest value of P? (, ) 3 0 B. maimize; 2 0 Graph each inequalit on the grid. Shade the feasible region in which the inequalities overlap. What are the four vertices of the feasible region? 5 5 (0, ), (, 2), (, ), and (, ) Tr each point in the objective function. Which verte results in the greatest value of P? Verte P 3 4 (0, 0) P 3(0) 4(0) 0 P 3(0) 4(2) 8 (, ) Copright b Holt, Rinehart and Winston. 47 Holt Algebra 2
Read To Go On? Problem Solving Intervention 3-4 Linear Programming Trail mi is available in Package A and Package B, as shown in the table. You want to have at least 24 ounces of nuts and at least 16 ounces of dried fruit. How man of each package should ou bu to minimize the cost? Nuts Dried Fruit Cost per package Package A 4 oz 2 oz $4 Package B 6 oz 5 oz $12 Understand the Problem 1. What is trail mi made of? 2. What does to minimize cost mean? Make a Plan 3. Let be the number of packages of A and let be the number of packages of. 4. Constraint Inequalit Graph the feasible region. ounces of nuts: 6 5 ounces of : 2 Number of Package A: Number of Package B: 0 2 10 3 Solve 5. Find the vertices of the feasible region b finding where the boundar lines intersect. 4 6 24 and 0 meet at the point. 2 5 16 and 0 meet at the point. 4 6 24 and 2 5 16 meet at the point. 6. The total cost P is given b the cost equation P 4 12. Evaluate the cost equation at each verte of the feasible region. For (0, 4): P 4 12 4 12(4) 0 $ For (8, 0): P 4 12 4 12(0) 0 $ For (3, 2): P 4 12 (3) 12 $ 7. The smallest value,, is with packages of A and packages of B. Look Back 8. Does our answer in Eercise 9 meet the goal in the problem statement? Copright b Holt, Rinehart and Winston. 48 Holt Algebra 2
Read To Go On? Quiz 3-1 Using Graphs and Tables to Solve Linear Sstems Solve each sstem b using a graph and a table. Check our answer. 1. 2 3 2. 4 3 1 3 3. 1 3 7 3 5 5 5 8 6 4 2 2 2 4 6 5 8 Classif each sstem, and determine the number of solutions. 4. 1 2 3 5. 2 6 2 8 10 4 3 6. 4 3 2 5 1 3-2 Using Algebraic Methods to Solve Linear Sstems Use substitution to solve each sstem of equations. 7. 1 8. 2 7 4 9. 3 2 18 3 2 5 Use elimination to solve each sstem of equations. 10. 3 2 0 11. 2 8 3 4 10 5 2 18 12. 7 2 5 Copright b Holt, Rinehart and Winston. 49 Holt Algebra 2
Read To Go On? Quiz continued 3-3 Solving Sstems of Linear Inequalities Graph each sstem of inequalities. 13. 8 14. 8 1 6 15. 8 0 10 10 5 10 10 10 5 5 10 10 16. There are 30 seats on the tour bus. A child s ticket costs $3 and an adult s ticket cost $5.50. The bus compan needs at least $120 to earn a profit from each tour. Write and graph a sstem of inequalities that can be used to determine the number of adults a and children c needed to make a profit. 3-4 Linear Programming 10 20 30 40 50 Graph each feasible region, and maimize or minimize the objective function P 2 3. 2 4 5 2 1 5 17. minimize; 2 18. maimize; 2 0 0 5 50 40 30 20 10 a 5 c 19. Samantha wants to add at least 40 fish to her new tank. She cannot use more than 25 of Fish A or more than 30 of Fish B. Fish A cost $5 each and Fish B cost $3 each. How man of each fish should she use in order to minimize the cost? Copright b Holt, Rinehart and Winston. 50 Holt Algebra 2
Read To Go On? Enrichment Linear Sstems in Two Dimensions You alread know how to graph an equation. If ou limit the domain or the range for the equation, ou can show line segments, letters, and even words or pictures. Graph 9 Graph 4 Graph 5 from 1 to 5. from 9 to 5. from 1 to 5. 10 8 6 4 2 5 4 3 2 1 10 8 6 4 2 5 4 3 2 1 10 8 6 4 2 5 4 3 2 1 1. Put the three parts together on one coordinate grid. What does it look like? 5 10 5 10 2. Cop the lines from Eercise 1 onto the coordinate grid below. Then draw lines on the coordinate grid for each set of equations below. a. 4 from 1 to 5, b. c. 5 5 from 4 to 1, from 1 to 5, from 6 to 10, 3 from 4 to 1, 6 8 1 from 4 to 1 from 1 to 5 from 1 to 5 5 10 5 10 Copright b Holt, Rinehart and Winston. 51 Holt Algebra 2
3B Find these vocabular words in Lesson 3-5 and the Multilingual Glossar. Vocabular Read To Go On? Skills Intervention 3-5 Linear Equations in Three Dimensions three-dimensional coordinate sstem ordered triple z-ais Graphing Points in Three Dimensions Graph ( 2, 1, 3) in three-dimensional space. Which value describes the -ais? z Should ou move forward or back on the -ais? Move back units on the -ais. Which value describes the -ais? Should ou move left or right on the -ais? Move 1 unit on the -ais. Which value describes the z-ais? Should ou move up or down on the z-ais? Move 3 units on the z-ais. Graphing Linear Equations in Three-Dimension Graph 2 z 2 in three-dimensional space. Find the intercepts b substituting in zero. z -intercept: 2(0) 2(0) 2 So. Plot (, 0, 0). -intercept: (0) 2 (0) 2 So. Plot (0,, 0). z-intercept: (0) 2(0) z 2 So z. Plot (0, 0, ). Connect each point. Copright b Holt, Rinehart and Winston. 52 Holt Algebra 2
3B Read To Go On? Problem Solving Intervention 3-5 Linear Equations in Three Dimensions Each point in coordinate space can be represented b an ordered triple of the form (,, z ). A coffee shop sells small, medium, and large coffee for $2, $3, and $4, respectivel. The want to make $210 in sales. The table shows some of the combinations of small, medium, and large coffees that result in a sales total of $210. Write a linear equation in three variables to represent the situation. Then complete the table for the possible numbers of small, medium, and large coffees. small medium large black 20 30? cream? 20 20 sugar 15? 30 cream and sugar 20 30? Understand the Problem 1. Define the variables: Let represent the number of small coffees, let represent the number of medium coffees, and let z represent the number of coffees. Then the income from small coffees is, the income from is 3, and the income from large coffee is. Make a Plan 2. Write an epression in three variables for the income. 3 Solve 3. Set the epression from Eercise 2 equal to the amount of total sales. 3 4. Use the equation to find each missing cell in the table: Row Equation Solve 1 2(20) 3(30) 4z 40 4z ; z 2 3(20) 4(20) 210 60 210; 3 2(15) (30) 210 3 210; 4 (20) 3(30) 4z 210; z = Look Back 5. Check the equation for each line of the table. Substitute the values in the table back into the equation 2 3 4z 210 to check. For eample, row 1: 2(20) 3(30) 4(20) 40 90 80 Copright b Holt, Rinehart and Winston. 53 Holt Algebra 2
3B Read To Go On? Skills Intervention 3-6 Solving Linear Sstems in Three Variables Solving a Linear Sstem in Three Variables 2 3 3z 9 Use elimination to solve 6 2 z 0. 4 2z 0 STEP 1 Eliminate one variable so instead of a 3-b-3 sstem, ou will have a 2-b-2 sstem. Look for the variable that is easiest to eliminate. In this sstem, eliminate. First, multipl equation 1 b 3 and then add it to equation 2. 3(2 3 3z 9) 9z Multipl the first equation b 3. 9z 6 2 z 0 Add equations 1 and 2. 10 Use this equation for the 2-b-2 sstem. Now, eliminate from equation 3 b multipling equation 1 b 2. Then add it to equation 3. 2(2 3 3z 9) 6 6z Multipl the first equation. 6 6z 4 2z 0 Add equations 1 and 3. 8z 10z 27 Write the 2-b-2 sstem: 7 z 18 Use this equation for the 2-b-2 sstem. STEP 2 Eliminate another variable; in this case,. Then solve for z. 7( 11 10z 27) z 189 Multipl the first equation b 7 and 11(7 8z 18) z 198 the second b 11. Then add and solve for z. 7 8z 18 7 8 4 2z 0 4 2 z 9 z What is the final solution? (,, ) 18 STEP 3 Substitute z into one of the equations from the 2-b-2 sstem to solve for. 0 STEP 4 Substitute and z into one of the original equations to solve for. Copright b Holt, Rinehart and Winston. 54 Holt Algebra 2
3B Read To Go On? Problem Solving Intervention 3-6 Solving Linear Sstems in Three Variables Part of Hannah s training program, 3 das per week, is to jog, walk, and run for various lengths of time. This table shows her training schedule. Write a sstem in three variables to represent the data in the table. What is Hannah s eercise rate for jogging, walking, and running? Da Jog Walk Run Total Distance Mon 3 h 2 h 1 h 29 mi Wed 4 h 3 h 1 h 37 mi Fri 2 h 1 h 2 h 29 mi Understand the Problem 1. What three eercises make up Hannah s training program? 2. What are the units for rate in this problem? Make a Plan 3. Let be Hannah s jogging rate in miles per hour, let be her, and z be her rate. Complete: Time Rate Solve 3 2 z 29 5. Write a sstem of equations to represent the data. 3 37 2 6. Using Equation 2 and Equation 1, eliminate the z-variable. Then, using Equation 2 and Equation 3, eliminate the z-variable to get a 2-b-2 sstem. 2 z 29 Multipl b 1. 6 2z Multipl b 2. 4 3 z 37 2 2z 29 Add. 5 Add. Rewrite as a 2-b-2 sstem: 8 Solve for. 8 6 5 45 6 5 45 Substitute for. 6 5(8 ) 45 6 40 5 45 ; So, 8 Now, find z. 2 2z 29 2( ) 2z 29 z 7. Hannah jogs at a rate of, she walks at a rate of, and she runs at a rate of. Look Back 8. Check the rates b substituting the values for the variables into each equation. Does our solution check? Copright b Holt, Rinehart and Winston. 55 Holt Algebra 2
3B Read To Go On? Quiz 3-5 Linear Equations in Three Dimensions Graph each point in three-dimensional space. 1. ( 5, 1, 4) 2. ( 4, 3, 4) 3. (4, 2, 2) z z z Graph each linear equation in three-dimensional space. 4. 3 12 2z 12 5. 2 2 4z 8 6. 9 9 3z 18 z z z Use the following information and the table for Eercises 7 and 8. An auto detailer charges $6 for a basic car wash, $10 to wa a car, and $15 for interior cleaning. The auto detailer s income was eactl $450 for each of the das shown in the table. Da Basic wash Wa Interior Monda 25 18 Tuesda 15 6 Wednesda 18 10 7. Write a linear equation in three variables to represent this situation. 8. Complete the table for the possible numbers of cleanings each da. Copright b Holt, Rinehart and Winston. 56 Holt Algebra 2
3B Read To Go On? Quiz continued 3-6 Solving Linear Sstems in Three Variables Use elimination to solve each sstem of equations. 2 3z 7 9. 2 z 4 3 2 2z 10 2 z 10 10. 3 2z 8 4 2 3z 10 2 3z 9 11. 2 2z 9 z 6 Use the following information and the table for Eercises 12 and 13. A pizza stand sells three different tpes of pizza: cheese, pepperoni, and vegetable. The table shows the total revenue for three hours on a particular afternoon. Time Cheese Pepperoni Vegetable Revenue 11:00 A.M. 12:00 P.M. 15 20 10 $95 12:00 P.M. 1:00 P.M. 18 16 12 $94 1:00 P.M. 2:00 P.M. 14 10 8 $64 12. Write a sstem in three variables to represent the data in the table. 13. How much does each tpe of slice of pizza cost? Cheese $, Pepperoni $, Vegetable $ Classif each sstem as consistent or inconsistent, and determine the number of solutions. 3 z 1 14. 2 2z 2 2 3z 1 3 2 z 0 15. 6 3z 4 2 8 2z 4 2 5 5z 27 16. 3 5z 4 2 3z 9 Copright b Holt, Rinehart and Winston. 57 Holt Algebra 2
3B Read To Go On? Enrichment Linear Sstems in Two Dimensions The formula for the distance, d, between two points ( 1, 1 ) and ( 2, 2 ) in a twodimensional coordinate plane is d ( 1 2 ) 2 ( 1 2 ) 2. There is a similar formula for the distance, d, between two points in threedimensional space. If ( 1, 1, z 1 ) and ( 2, 2, z 2 ) are two points in threedimensional space, then the formula for the distance, d, between the points is d ( 1 2 ) 2 ( 1 2 ) 2 (z 1 z 2 ) 2. For eample, the distance, d, between (3, 5, 2) and (6, 2, 4) is: d (3 6) 2 (5 ( 2)) 2 ( 2 4 ) 2 ( 3) 2 (7) 2 ( 6) 2 9 49 36 94 or about 9.7 units. Find the distance between each pair of points in three-dimensional space. 1. (1, 2, 3) and (4, 5, 6) 2. (0, 0, 0) and (2, 2, 2) 3. (5, 2, 6) and (5, 2, 9) 4. (3, 8, 1) and (3, 5, 1) 5. Look at Eercises 3 and 4. In each pair of points, how are the coordinates the same and how are the different? Can ou relate that to the distance between each pair of points? 6. A bo is 12 centimeters long, 8 centimeters deep, and 3 centimeters tall. Calculate the length of the longest rod that can fit in the bo. a. Use (0, 0, 0) as the coordinates of one corner of the bo. What are the coordinates of the opposite corner of the bo? b. What is the length of the bo from one corner to the opposite corner? 7. What is the length of the longest rod that can fit in each bo with the given dimensions? a. 15 in. b 12 in. b 8 in. b. 20 cm b 10 cm b 2 cm c. a cube with edge 20 cm d. a cube with edge 24 in. Copright b Holt, Rinehart and Winston. 58 Holt Algebra 2