Maintaining Mathematical Proficiency

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1 Name Date Chapter 5 Maintaining Mathematical Proficienc Graph the equation =. = = = 5. 3 = = 1 Solve the inequalit. Graph the solution. 7. a 3 > 8. c 9. d 5 < r 5 r Copright Big Ideas Learning, LLC Algebra 1 131

2 Name Date 5.1 Solving Sstems of Linear Equations b Graphing For use with Eploration 5.1 Essential Question How can ou solve a sstem of linear equations? 1 EXPLORATION: Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast. The spend \$600 preparing a bedroom to rent. The cost to our famil for food and utilities is \$15 per night. The charge \$75 per night to rent the bedroom. a. Write an equation that represents the costs. Cost, C \$15 per Number of = + \$600 in dollars night nights, ( ) b. Write an equation that represents the revenue (income). Revenue, R \$75 per Number of = in dollars night nights, ( ) c. A set of two (or more) linear equations is called a sstem of linear equations. Write the sstem of linear equations for this problem. EXPLORATION: Using a Table or Graph to Solve a Sstem Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Use the cost and revenue equations from Eploration 1 to determine how man nights our famil needs to rent the bedroom before recovering the cost of preparing the bedroom. This is the break-even point. a. Complete the table. (nights) C (dollars) R (dollars) 13 Algebra 1 Copright Big Ideas Learning, LLC

3 Name Date 5.1 Solving Sstems of Linear Equations b Graphing (continued) EXPLORATION: Using a Table or Graph to Solve a Sstem (continued) b. How man nights does our famil need to rent the bedroom before breaking even? c. In the same coordinate plane, graph the cost equation and the revenue equation from Eploration d. Find the point of intersection of the two graphs. What does this point represent? How does this compare to the break-even point in part (b)? Eplain. Communicate Your Answer 3. How can ou solve a sstem of linear equations? How can ou check our solution?. Solve each sstem b using a table or sketching a graph. Eplain wh ou chose each method. Use a graphing calculator to check each solution. a. = = b. = = c. = 1 = Copright Big Ideas Learning, LLC Algebra 1 133

4 Name Date 5.1 Notetaking with Vocabular For use after Lesson 5.1 In our own words, write the meaning of each vocabular term. sstem of linear equations solution of a sstem of linear equations Core Concepts Solving a Sstem of Linear Equations b Graphing Step 1 Graph each equation in the same coordinate plane. Step Estimate the point of intersection. Step 3 Check the point from Step b substituting for and in each equation of the original sstem. Notes: 13 Algebra 1 Copright Big Ideas Learning, LLC

5 Name Date 5.1 Notetaking with Vocabular (continued) Etra Practice In Eercises 1 6, tell whether the ordered pair is a solution of the sstem of linear equations. 1. (3,1); + = = 3. (1, 3); = + = 5 3. (, 0); = = ( 1, ); = 3 = 0 5. (,3);3 = 1 + = 9 6. (, 3); + = 3 3 = 1 In Eercises 7 9, use the graph to solve the sstem of linear equations. Check our solution = 10 + = 0 8. = 5 + = = 8 3 = Copright Big Ideas Learning, LLC Algebra 1 135

6 Name Date 5.1 Notetaking with Vocabular (continued) In Eercises 10 15, solve the sstem of linear equations b graphing. 10. = = = 6 = + 5 = 1 = = = = 1 + = = = A test has twent questions worth 100 points. The test consists of true-false questions worth points each and multiple choice questions worth 8 points each. How man of each tpe of question are on the test? Algebra 1 Copright Big Ideas Learning, LLC

7 Name Date 5. Solving Sstems of Linear Equations b Substitution For use with Eploration 5. Essential Question How can ou use substitution to solve a sstem of linear equations? 1 EXPLORATION: Using Substitution to Solve Sstems Work with a partner. Solve each sstem of linear equations using two methods. Method 1 Solve for first. Solve for in one of the equations. Substitute the epression for into the other equation to find. Then substitute the value of into one of the original equations to find. Method Solve for first. Solve for in one of the equations. Substitute the epression for into the other equation to find. Then substitute the value of into one of the original equations to find. Is the solution the same using both methods? Eplain which method ou would prefer to use for each sstem. a. + = = 5 b. 6 = = 7 c. + = = 18 Copright Big Ideas Learning, LLC Algebra 1 137

8 Name Date 5. Solving Sstems of Linear Equations b Substitution (continued) EXPLORATION: Writing and Solving a Sstem of Equations Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. a. Write a random ordered pair with integer coordinates. One wa to do this is to use a graphing calculator. The ordered pair, 3. generated at the right is ( ) Choose two random integers between 5 and 5. randint(-5 5 ) {- -3} b. Write a sstem of linear equations that has our ordered pair as its solution. c. Echange sstems with our partner and use one of the methods from Eploration 1 to solve the sstem. Eplain our choice of method. Communicate Your Answer 3. How can ou use substitution to solve a sstem of linear equations?. Use one of the methods from Eploration 1 to solve each sstem of linear equations. Eplain our choice of method. Check our solutions. a. + = 7 = 9 b. = 6 + = c. 3 + = 10 + = 6 d. 3 + = 13 3 = 3 e. 3 = 9 3 = 8 f. 3 = = Algebra 1 Copright Big Ideas Learning, LLC

9 Name Date 5. Notetaking with Vocabular For use after Lesson 5. In our own words, write the meaning of each vocabular term. sstem of linear equations solution of a sstem of linear equations Core Concepts Solving a Sstem of Linear Equations b Substitution Step 1 Solve one of the equations for one of the variables. Step Substitute the epression from Step 1 into the other equation and solve for the other variable. Step 3 Substitute the value from Step into one of the original equations and solve. Notes: Copright Big Ideas Learning, LLC Algebra 1 139

10 Name Date 5. Notetaking with Vocabular (continued) Etra Practice In Eercises 1 18, solve the sstem of linear equations b substitution. Check our solution = 10 = 5. = 3 = = 1 =. = 3 = = 3 = = 5 = = + 8 = 8. = = = 0 = 10 Algebra 1 Copright Big Ideas Learning, LLC

11 Name Date 5. Notetaking with Vocabular (continued) = 3 + = = = = 3 = = 8 5 = 1. = = = = = = = + = = = An adult ticket to a museum costs \$3 more than a children s ticket. When 00 adult tickets and 100 children s tickets are sold, the total revenue is \$100. What is the cost of a children s ticket? Copright Big Ideas Learning, LLC Algebra 1 11

12 Name Date 5.3 Solving Sstems of Linear Equations b Elimination For use with Eploration 5.3 Essential Question How can ou use elimination to solve a sstem of linear equations? 1 EXPLORATION: Writing and Solving a Sstem of Equations Work with a partner. You purchase a drink and a sandwich for \$.50. Your friend purchases a drink and five sandwiches for \$ You want to determine the price of a drink and the price of a sandwich. a. Let represent the price (in dollars) of one drink. Let represent the price (in dollars) of one sandwich. Write a sstem of equations for the situation. Use the following verbal model. Number Price Number of Price per Total + = of drinks per drink sandwiches sandwich price Label one of the equations Equation 1 and the other equation Equation. b. Subtract Equation 1 from Equation. Eplain how ou can use the result to solve the sstem of equations. Then find and interpret the solution. EXPLORATION: Using Elimination to Solve Sstems Work with a partner. Solve each sstem of linear equations using two methods. Method 1 Method Subtract. Subtract Equation from Equation 1.Then use the result to solve the sstem. Add. Add the two equations. Then use the result to solve the sstem. Is the solution the same using both methods? Which method do ou prefer? a. 3 = = 0 b. + = 6 = c. = 7 + = 5 1 Algebra 1 Copright Big Ideas Learning, LLC

13 Name Date 5.3 Solving Sstems of Linear Equations b Elimination (continued) 3 EXPLORATION: Using Elimination to Solve a Sstem Work with a partner. + = 7 Equation = 17 Equation a. Can ou eliminate a variable b adding or subtracting the equations as the are? If not, what do ou need to do to one or both equations so that ou can? b. Solve the sstem individuall. Then echange solutions with our partner and compare and check the solutions. Communicate Your Answer. How can ou use elimination to solve a sstem of linear equations? 5. When can ou add or subtract the equations in a sstem to solve the sstem? When do ou have to multipl first? Justif our answers with eamples. 6. In Eploration 3, wh can ou multipl an equation in the sstem b a constant and not change the solution of the sstem? Eplain our reasoning. Copright Big Ideas Learning, LLC Algebra 1 13

14 Name Date 5.3 Notetaking with Vocabular For use after Lesson 5.3 In our own words, write the meaning of each vocabular term. coefficient Core Concepts Solving a Sstem of Linear Equations b Elimination Step 1 Multipl, if necessar, one or both equations b a constant so at least one pair of like terms has the same or opposite coefficients. Step Add or subtract the equations to eliminate one of the variables. Step 3 Solve the resulting equation. Step Substitute the value from Step 3 into one of the original equations and solve for the other variable. Notes: 1 Algebra 1 Copright Big Ideas Learning, LLC

15 Name Date 5.3 Notetaking with Vocabular (continued) Etra Practice In Eercises 1 18, solve the sstem of linear equations b elimination. Check our solution = 17 + = 8. = = = 10 =. + 3 = 6 3 = = = = = = = = 1 = = + = 1 Copright Big Ideas Learning, LLC Algebra 1 15

16 Name Date 5.3 Notetaking with Vocabular (continued) = 0 + = = 3 = = = = = = 7 3 = = = = 7 5 = = = = = The sum of two numbers is. The difference is 6. What are the two numbers? 16 Algebra 1 Copright Big Ideas Learning, LLC

18 Name Date 5. Solving Special Sstems of Linear Equations (continued) Communicate Your Answer 3. Can a sstem of linear equations have no solution or infinitel man solutions? Give eamples to support our answers.. Does the sstem of linear equations represented b each graph have no solution, one solution, or infinitel man solutions? Eplain. a. b. c. = + = = = 3 + = = 18 Algebra 1 Copright Big Ideas Learning, LLC

19 Name Date 5. Notetaking with Vocabular For use after Lesson 5. In our own words, write the meaning of each vocabular term. parallel Core Concepts Solutions of Sstems of Linear Equations A sstem of linear equations can have one solution, no solution, or infinitel man solutions. One solution No solution Infinitel man solutions Notes: The lines intersect. The lines are parallel. The lines are the same. Copright Big Ideas Learning, LLC Algebra 1 19

20 Name Date 5. Notetaking with Vocabular (continued) Etra Practice In Eercises 1 18, solve the sstem of linear equations. 1. = 3 7. = 5 1 = 3 + = = = = 6 3 = = = = = = = = = = = Algebra 1 Copright Big Ideas Learning, LLC

21 Name Date 5. Notetaking with Vocabular (continued) = 1 5 = = = = 3 + = = 9 5 = = = = = = = = + 3 = = = You have \$15 in savings. Your friend has \$5 in savings. You both start saving \$5 per week. Write a sstem of linear equations that represents this situation. Will ou ever have the same amount of savings as our friend? Eplain. Copright Big Ideas Learning, LLC Algebra 1 151

22 Name Date 5.5 Solving Equations b Graphing For use with Eploration 5.5 Essential Question How can ou use a sstem of linear equations to solve an equation with variables on both sides? 1 EXPLORATION: Solving an Equation b Graphing Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Solve 1 = 1 + b graphing. a. Use the left side to write a linear equation. Then use the right side to write another linear equation. b. Graph the two linear equations from part (a). Find the -value of the point of intersection. Check that the -value is the solution of 1 = c. Eplain wh this graphical method works. EXPLORATION: Solving Equations Algebraicall and Graphicall Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Solve each equation using two methods. Method 1 Method Use an algebraic method. Use a graphical method. Is the solution the same using both methods? a. 1 + = b. + = Algebra 1 Copright Big Ideas Learning, LLC

23 Name Date 5.5 Solving Equations b Graphing (continued) EXPLORATION: Solving Equations Algebraicall and Graphicall (continued) c. 1 = 1 d. + 7 = e. +.5 = 0.5 f = Communicate Your Answer 3. How can ou use a sstem of linear equations to solve an equation with variables on both sides?. Compare the algebraic method and the graphical method for solving a linear equation with variables on both sides. Describe the advantages and disadvantages of each method. Copright Big Ideas Learning, LLC Algebra 1 153

24 Name Date 5.5 Notetaking with Vocabular For use after Lesson 5.5 In our own words, write the meaning of each vocabular term. absolute value equation Core Concepts Solving Linear Equations b Graphing Step 1 To solve the equation a + b = c + d, write two linear equations. = a + b a + b = c + d and = c + d Step Graph the sstem of linear equations. The -value of the solution of the sstem of linear equations is the solution of the equation a + b = c + d. Notes: 15 Algebra 1 Copright Big Ideas Learning, LLC

25 Name Date 5.5 Notetaking with Vocabular (continued) Etra Practice In Eercises 1 9, solve the equation b graphing. Check our solution(s) = = =. = 3( ) = = 7( ) Copright Big Ideas Learning, LLC Algebra 1 155

26 Name Date 5.5 Notetaking with Vocabular (continued) 7. 3 = = = 156 Algebra 1 Copright Big Ideas Learning, LLC

27 Name Date 5.6 Graphing Linear Inequalities in Two Variables For use with Eploration 5.6 Essential Question How can ou graph a linear inequalit in two variables? A solution of a linear inequalit in two variables is an ordered pair (, ) that makes the inequalit true. The graph of a linear inequalit in two variables shows all the solutions of the inequalit in a coordinate plane. 1 EXPLORATION: Writing a Linear Inequalit in Two Variables Work with a partner. a. Write an equation represented b the dashed line. b. The solutions of an inequalit are represented b the shaded region. In words, describe the solutions of the inequalit. c. Write an inequalit represented b the graph. Which inequalit smbol did ou use? Eplain our reasoning. EXPLORATION: Using a Graphing Calculator Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Use a graphing calculator to graph 1 3. a. Enter the equation = 1 3 into our calculator b. The inequalit has the smbol. So, the region to be shaded is above the graph of = 1 3, as shown. Verif this b testing a point in this region, such as (0, 0), to make sure it is a solution of the inequalit Because the inequalit smbol is greater than or equal to, the line is solid and not dashed. Some graphing calculators alwas use a solid line when graphing inequalities. In this case, ou have to determine whether the line should be solid or dashed, based on the inequalit smbol used in the original inequalit. Copright Big Ideas Learning, LLC Algebra 1 157

28 Name Date 5.6 Graphing Linear Inequalities in Two Variables (continued) 3 EXPLORATION: Graphing Linear Inequalities in Two Variables Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Graph each linear inequalit in two variables. Eplain our steps. Use a graphing calculator to check our graphs. a. > + 5 b c Communicate Your Answer. How can ou graph a linear inequalit in two variables? 5. Give an eample of a real-life situation that can be modeled using a linear inequalit in two variables. 158 Algebra 1 Copright Big Ideas Learning, LLC

29 Name Date 5.6 Notetaking with Vocabular For use after Lesson 5.6 In our own words, write the meaning of each vocabular term. linear inequalit in two variables solution of a linear inequalit in two variables graph of a linear inequalit half-planes Core Concepts Graphing a Linear Inequalit in Two Variables Step 1 Graph the boundar line for the inequalit. Use a dashed line for < or >. Use a solid line for or. Step Test a point that is not on the boundar line to determine whether it is a solution of the inequalit. Step 3 When a test point is a solution, shade the half-plane that contains the point. When the test point is not a solution, shade the half-plane that does not contain the point. Notes: Copright Big Ideas Learning, LLC Algebra 1 159

30 Name Date 5.6 Notetaking with Vocabular (continued) Etra Practice In Eercises 1 6, tell whether the ordered pair is a solution of the inequalit > 5; ( 3, ). ; ( 5, 3) 3. + ; ( 1,). 5 + < 7; (, ) 5. 3 > 6; ( 1, 1) 6. 5; (, 3) In Eercises 7 18, graph the inequalit in a coordinate plane. 7. < 8. > 1 9. > < 1. > 160 Algebra 1 Copright Big Ideas Learning, LLC

31 Name Date 5.6 Notetaking with Vocabular (continued) 13. < < < > An online store sells digital cameras and cell phones. The store makes a \$100 profit on the sale of each digital camera and a \$50 profit on the sale of each cell phone. The store wants to make a profit of at least \$300 from its sales of digital cameras and cell phones. Write and graph an inequalit that represents how man digital cameras and cell phones the must sell. Identif and interpret two solutions of the inequalit Copright Big Ideas Learning, LLC Algebra 1 161

32 Name Date 5.7 Sstems of Linear Inequalities For use with Eploration 5.7 Essential Question How can ou graph a sstem of linear inequalities? 1 EXPLORATION: Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Eplain our reasoning. + Inequalit 1 0 Inequalit A. B. EXPLORATION: Graphing a Sstem of Linear Inequalities Go to BigIdeasMath.com for an interactive tool to investigate this eploration. Work with a partner. Consider the linear inequalities given in Eploration 1. + Inequalit 1 0 Inequalit a. Use two different colors to graph the inequalities in the same coordinate plane. What is the result? 16 Algebra 1 Copright Big Ideas Learning, LLC

33 Name Date 5.7 Sstems of Linear Inequalities (continued) EXPLORATION: Graphing a Sstem of Linear Inequalities (continued) b. Describe each of the shaded regions of the graph. What does the unshaded region represent? Communicate Your Answer 3. How can ou graph a sstem of linear inequalities?. When graphing a sstem of linear inequalities, which region represents the solution of the sstem? 5. Do ou think all sstems of linear inequalities have a solution? Eplain our reasoning. 6. Write a sstem of linear inequalities represented b the graph. 6 Copright Big Ideas Learning, LLC Algebra 1 163

34 Name Date 5.7 Notetaking with Vocabular For use after Lesson 5.7 In our own words, write the meaning of each vocabular term. sstem of linear inequalities solution of a sstem of linear inequalities graph of a sstem of linear inequalities Core Concepts Graphing a Sstem of Linear Inequalities Step 1 Graph each inequalit in the same coordinate plane. Step Find the intersection of the half-planes that are solutions of the inequalities. This intersection is the graph of the sstem. < Notes: 16 Algebra 1 Copright Big Ideas Learning, LLC

35 Name Date 5.7 Notetaking with Vocabular (continued) Etra Practice In Eercises 1, tell whether the ordered pair is a solution of the sstem of linear inequalities. 0, 0 ; > 1. ( ) < 1, 1 ; < 3. ( ) > 3. (, 3); + +. (0, ); In Eercises 5 8, graph the sstem of linear inequalities. 5. > 3 6. < 3 < Copright Big Ideas Learning, LLC Algebra 1 165

36 Name Date 5.7 Notetaking with Vocabular (continued) < 6 < + 1 In Eercises 9 1, write a sstem of linear inequalities represented b the graph Algebra 1 Copright Big Ideas Learning, LLC

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