1 Practice Solving Sstems b Graphing Solve each sstem b graphing. Check our solution. 1. =- + 3 = - (1, ). = 1 - (, 1) = = = 1 (, 3). =-5 = - 7. = = 0 (1, 5) = 5 =-7 (, 7). =- - = + 9 same graph means 8. = 3-3 =- (3, 1) 9. =- 5 - infinitel =--5 man solutions ( 3, ) ( 5, 0) 10. Reasoning Can there be more than one point of intersection between the graphs of two linear equations? Wh or wh not? Unless the graphs of two linear equations coincide, there can be onl one point of intersection, because two lines can intersect in at most one point. 11. Reasoning If the graphs of the equations in a sstem of linear equations coincide with each other, what does that tell ou about the solution of the sstem? Eplain. If the graphs of two linear equations coincide, then there are infinitel man solutions to the sstem because ever solution of one equation is also a solution of the other equation. 1. Writing Eplain the method used to graph a line using the slope and -intercept. First use the -intercept to plot a point on the -ais. From that point, move one unit to the right and move verticall the value of the slope to plot a second point. Then connect the two points. 13. Reasoning If the ordered pair (3, -) satisfies one of the two linear equations in a sstem, how can ou tell whether the point satisfies the other equation of the sstem? Eplain. Substitute 3 for and for into the other equation. If the resulting equation is true, (3, ) is a solution to the equation. 1. Writing If the graphs of two lines in a sstem do not intersect at an point, what can ou conclude about the solution of the sstem? Wh? Eplain. If the lines do not intersect, there is no solution to the sstem because no ordered pair satisfies both equations. 15. Reasoning Without graphing, decide whether the following sstem of linear equations has one solution, infinitel man solutions or no solution. Eplain. = 3-5 = + 10 The sstem has infinitel man solutions because when ou rewrite the second equation in slope-intercept form, it is identical to the first equation. 1. Five ears from now, a father s age will be three times his son s age, and 5 ears ago, he was seven times as old as his son was. What are their present ages? father is 0; son is 10 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
2 Practice (continued) Solving Sstems b Graphing 17. The denominator of a fraction is greater than its numerator b 9. If 7 is subtracted from both its numerator and denominator, the new fraction equals 3. What is the original fraction? The sum of the distances two hikers walked is 53 mi, and the difference is 5 mi. What are the distances? 39 mi; 1 mi 19. The result of dividing a two-digit number b the number with its digits reversed is 7. If the sum of the digits is 1, what is the number? 8 Solve each sstem b graphing. Tell whether the sstem has one solution, infinitel man solutions, or no solution. 0. = =-3 1. = + 1 = = 8 = (, 1); one solution (1, 3); one solution (3, ); one solution 3. =- + 1 =-. = = 1 ( 3, 7); one solution no solution. = 1 - = = = (8, ); one solution no solution 5. = 5-15 = 3 + (, 5); one solution 8. =- - 7 = + 5 (, 3); one solution = 1 - =- + 7 infinitel man solutions 30. = =- (0, ); one solution 31. =- 3-3 = 1 + (, 3); one solution 3. The measure of one of the angles of a triangle is 35. The sum of the measures of the other two angles is 15 and the difference between their measures is 15. What are the measures of the unknown angles? 80 and 5 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
3 Practice Solving Sstems Using Substitution Solve each sstem b substitution. Check our solution. 1. = + = 3 (1, 1). =- + = 3 (1, 3) 3. = - 10 (, ) = - 8. = = 7 ( 3, 1) 5. + = 1 = 3-1 (, ). - 3 = 13 (5, 1) = = = 7.5 (.5, ) 8. - = 5 (7, 3) 9. = - - (, 7) -9 + = = Writing How do ou know that substitution gives the answer to a sstem of equations? Eplain. You can verif our answer b substituting the - and - values into the original equations. 11. Reasoning With the substitution method, which variable should ou solve for first? Eplain. You should solve for a variable that alread has a coefficient of 1 or Writing How can ou use substitution method to solve a sstem of equations that does not have a variable with a coefficient of 1 or -1? You can solve for a variable b isolating that term and then dividing b the coefficient. + = Writing When solving the sstem of equations + = 8 using substitution, which variable will ou solve for and which equation will ou use to substitute into? You would solve the second equation for and then substitute back into the first equation. 1. Reasoning Can ou tell that there is no solution for a sstem b just looking at the equations? Eplain and give an eample. If all variables of one equation are multiples of the other, then there is no solution to the sstem. 15. If the difference in the side lengths of two squares is 10, and the sum of the side lengths is 18, what are the side lengths? 1 and 1. A shopper purchased 8 T-Shirts and 5 pairs pants for $0. The net da, he purchased 5 T-shirts and 1 pair of pants for $11. How much does each T-shirt and each pair of pants cost? t-shirts: $0; pants: $1 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
4 Practice (continued) Solving Sstems Using Substitution 17. A student bought 1 bo of craons and 5 reams of paper for $5. She bought 5 boes of craons and 3 reams of paper for $50. What is the cost of each bo of craons and each ream of paper? craons: $; paper: $ Suppose ou got 8 mangoes and 3 apples for $18 and 3 mangoes and 5 apples for $1.50. How much does each mango and each apple cost? mango: $1.50; apple: $ A shopper purchased tables and chairs for $00 and tables and 7 chairs for $00. What is the cost of each table and each chair? tables: $5; chairs: $50 0. If the length of the rectangle is twice the width, and the perimeter of the rectangle is 30 cm, what is length and width of the rectangle? 10 cm; 5 cm 1. The population of a cit is,500. If the number of males is 0 more than the number of females, how man males and females are there in the cit? 1130 females and 1370 males Solve each sstem b substitution. Tell whether the sstem has one solution, infinitel man solutions, or no solution = =-3 ( 3, ); one solution = = 0 (3, 1); one solution = = 7 (, 1); one solution =-10 + =-5 ( 1, 1); one solution. = = ( 5, 3); one solution 9. = = 10 ( 3, 1); one solution. = + 1 = - 5 no solution = 18 3 = infinitel man solutions = 7 + = ( 1, 3); one solution 31. = = = = = = 5 (, 3); one solution (, ); one solution (, 1); one solution Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
5 Practice Solving Sstems Using Elimination Solve each sstem using elimination = - = (3, 1). + = 3 (5, 1) 3. - = (, 8) - = 3 - =. - =- - + = 3 (, 1) = = 5 (, 1 3 ). + =- + = (8, ) = 8 - = 5 (3, 1 ) 8. - = = 9. - =- a 1 5, 7 5 b (1, ) - = = Writing For the sstem, which variable should ou eliminate + = 3 first and wh? How will ou eliminate that variable? You should eliminate the first because ou onl need to multipl one equation b a constant. You would multipl the second equation b 5 and then add the equations together to eliminate. 11. Open-Ended If ou do not have equal coefficients for both variables, can ou still use the elimination method? Eplain. You can use the elimination method b multipling one or both equations b a constant. 1. In a class, 5 students take the SAT eam. The number of bos is 8 more than the number of girls. a. Write a sstem that models the above situation. + = 5, = 8 b. Do ou need to multipl an of the equations b a constant? If so, which equation and what is the constant? You do not need to multipl either equation b a constant. 13. Open-Ended Write a sstem for which using the elimination method to solve the sstem is easier than the substitution method. Eplain. Check student s work. 1. Error Analsis A student solved a sstem of linear equations using the elimination method as follows. Describe and correct the error made b the student. 3-5 = - 10 = 8 Multipl equation 1 b = = Multipl equation b = 1 Add the equations. =- Divide b -7. When the student multiplied the second equation b 3, the student forgot to multipl the -term. Correct answer: (, 1) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
6 Practice (continued) Solving Sstems Using Elimination 15. A farm raises a total of 0 chickens and pigs. The number of legs of the stock in the farm totals 50. How man chickens and pigs are at the farm? 180 chickens and 0 pigs 1. You drive a car that runs on ethanol and gas. You have a 0-gallon tank to fill and ou can bu fuel that is either 5 percent ethanol or 85 percent ethanol. How much of each tpe of fuel should ou bu to fill our tank so that it is 50 percent ethanol? gal of 85% ethanol; 11 gal of 5% ethanol Your math test has 38 questions and is worth 00 points. The test consists of multiple-choice questions worth points each and open-ended questions worth 0 points each. How man of each tpe of question are there? 35 multiple choice; 3 open-ended 18. A student bought 3 boes of pencils and boes of pens for $. He then bought boes of pencils and boes of pens for $8. Find the cost of each bo of pencils and each bo of pens. pencils: $1.00; pens: $1.50 Solve each sstem using elimination. Tell whether the sstem has one solution, infinitel man solutions, or no solution =-7 = - 1 infinitel man solutions. 5 + = 15 3 =-15 + no solution 5. + = = (, 1); one solution = =-8 (, 1); one solution =- + 3 = (1, 1); one solution 3. 3 = = infinitel man solutions. + =- - - = no solution = 1 + = 10 (, 1); one solution 1. + = - =-1 (0, 3); one solution. 3 - =- - + = 8 (1, 5); one solution =-3 = infinitel man solutions = 0 + = 19 no solution 31. How is the multiplication or division propert of equalit used in the elimination method? Are the properties alwas needed? Eplain. The multiplication or division propert of equalit is used when ou multipl or divide an equation b a constant to get another true equation. It is not necessar when a variable alread has opposite coefficients. Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
7 Practice Applications of Linear Sstems Solve each word problem 1. You have $000 to invest in two stock funds. The first fund pas 5% annual interest and the second account pas 9% annual interest. If after a ear ou have made $380 in interest, how much mone did ou invest in each account? $000 at 5%; $000 at 9%. During a sale at the local department store, ou bu three sweatshirts and two pairs of sweatpants for $ Later ou return to the same store and bu three more sweatshirts and four more pairs of sweatpants for $13. What is the sale price of each sweatshirt and each pair of sweatpants? sweatshirt: $1; sweatpants: $ The sum of two numbers is 7. The larger number is 3 more than the smaller number. What are the two numbers? (15, 1). One plane at 50 feet is ascending at a rate of 0 feet per minute, while another plane at 3800 feet is descending at a rate of 10 feet per minute. How long will it take the two planes to be at the same altitude? 0.5 min 5. The perimeter of a rectangle is in. and its length is 3 times its width. What are the length and the width of the rectangle? length: 9 in.; width: 3 in.. You are getting read to move and have asked some friends to help. For lunch, ou bu the following sandwiches at the local deli for $30: si tuna sandwiches and si turke sandwiches. Later at night, everone is hungr again and ou bu four tuna sandwiches and eight turke sandwiches for $30.0. What is the price of each sandwich? tuna: $.35; turke: $.5 7. You have a cable plan that costs $39 a month for a basic plan plus one movie channel. Your friend has the same basic plan plan plus two movie channels for $5.50. What is the basic plan charge that ou both pa? $ At an all-ou-can-eat barbeque fundraiser that ou are sponsoring, adults pa $ for a dinner and children pa $ for a dinner. 1 people attend and ou raise $118. What is the total number of adults and the total number of children attending? a. What is a sstem of equations that ou can use to solve this problem? + = 1 + = 118 b. What method would ou use to solve the sstem? Wh? I d use substitution because in the first equation the coefficients are alread 1. Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
8 Practice (continued) Applications of Linear Sstems Solve each sstem. Eplain wh ou chose the method ou used. 9. = = 7 ( 3, 1); substitution because there was a coefficient of = 5 (7, 3); elimination -9 + =-9 because no coefficients were 1 or = 8 - = = 13 = 1-7 a3, 1 b ; elimination because the -coefficients are the same. (5, 1); substitution because is isolated in the second equation = 3 - = =-3a, 1 b ; elimination because = 5 the -coefficients are opposites 15. Open-Ended What are three differences between an inconsistent sstem and a consistent and independent sstem? Eplain. Answers ma var. Sample: There is one unique solution to an independent sstem, but no solution to an inconsistent sstem and infinitel man solutions to a dependent sstem. The graphs of the equations of an independent sstem have different slopes, but the slopes are the same for the lines of an inconsistent or dependent sstem. When solving a sstem algebraicall, ou get a specific value for each variable if the sstem is independent, but ou get a false statement for an inconsistent sstem and a statement that is alwas true for a dependent sstem. 1. Reasoning One number is less than 3 times a second number. If 3 more than two times the first number is decreased b two times the second, the result is 11. What are both numbers? (8, ) 17. Error Analsis In Eercise 1, what kind of errors are likel to occur when solving the problem? Answers ma var. Sample: You might misinterpret the description. For instance, ou might think less than 3 times a number means 3 instead of A plane leaves Chicago and flies 750 miles to New York. If it takes.5 hours to get to New York fling against the wind, but onl hours to fl back to Chicago, what is the plane s rate of speed and what is the wind speed? wind: 37.5 mi/h; plane: mi/h 19. A coin bank has 50 coins, dimes and quarters, worth $39.5. How man of each tpe of coin are there? 155 dimes; 95 quarters 0. In ears, a mother will be 5 times as old as her daughter. At present, the mother is 9 times as old as the daughter. How old are the mother and the daughter toda? mother: 3; daughter: (, 8); elimination because the -coefficients were the same. Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
9 Practice Linear Inequalities Graph each linear inequalit. 1. Ú Ú O O O O O HSM11A1TR_005_ANT003 O O 1 O 1 HSM11A1TR_005_ANT O Error Analsis A student graphed as shown. Describe and correct HSM11A1TR_005_ANT01 the student s error. The student shaded above the line when the student should have shaded below the line. 11. Writing How do ou decide which half-plane to shade when graphing an inequalit? Eplain. If the inequalit is of the form + m + b or # m + b, shade above the line. Otherwise, shade below the line. 8 8 O 8 8 HSM11A1TR_005_ANT00 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
10 Practice (continued) Linear Inequalities Determine whether the ordered pair is a solution of the linear inequalit , (-1, 1) , (, -1) not a solution solution , (, 1) ,(-8, -) solution solution 1. +, (-9, -5) , (3, 10) not a solution solution , (9, 1) , (8, 0.5) not a solution solution Write an inequalit that represents each graph (, ) (0, ) # (0, 5) 8 O 8 8 O 8 (3, ) # You and some friends have $30. You want to order large pizzas (p) that are $9 eachand drinks (d) that cost $1 each. Write and graph an inequalit that shows how man pizzas and drinks can ou order.? d " 9p Tickets to a pla cost $5 at the door and $ in advance. The theater club wants to raise at least $00 from the pla. Write and graph an inequalit for the number of tickets the theater club needs to sell. If the club sells 0 tickets in advance, how man do the need to sell at the door to reach their goal? 5 + # 00; 8 Pizza and Drinks for $30 p 8 10 Pizza. Reasoning Two students did a problem as above, but one used for the first variable and for the second variable and the other student used for the second variable and for the first variable. How did their answers differ and which one, if either, was incorrect? Drinks O O d The students answers are reversed. Neither one is incorrect if the variables are defined correctl. Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
11 Practice Sstems of Linear Inequalities Solve each sstem of inequalities b graphing O O O. Writing What is the difference between the solution of a sstem of linear inequalities and the solution of a sstem of linear equations? Eplain. The graph of a sstem of linear equations is made up of lines, so the solution, if it eists, is a single point. The graph of a sstem of linear inequalities is made up of shaded half-planes, so if a solution eists, it is a region where the shaded regions overlap. 5. Open-Ended When can ou sa that there is no solution for a sstem of linear inequalities? Eplain our answer and show with a sstem and graph. Answers ma var. Sample: There is no solution to a sstem of inequalities when the boundar lines are parallel and the shaded regions do not overlap. Eample: " 1; # + ; Check students graphs.. Error Analsis A student graphs the sstem below. Describe and correct the student s error. - Ú 3 - Ú 1 When graphing the inequalit # 1, the student shaded the wrong half of the plane. 8 8 O 8 8 Determine whether the ordered pair is a solution of the given sstem. 7. (0, 1); 1 - Ú (-, 3); (1, ); not a solution solution solution Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
12 Practice (continued) Sstems of Linear Inequalities 10. Mark is a student, and he can work for at most 0 hours a week. He needs to earn at least $75 to cover his weekl epenses. His dog-walking job pas $5 per hour and his job as a car wash attendant pas $ per hour. Write a sstem of inequalities to model the situation, and graph the inequalities. + " 0, + 5 # Britne wants to bake at most 10 loaves of bread for a bake sale. She wants to make banana bread that sells for $1.5 each and nut bread that sells for $1.50 each and make at least $ in sales. Write a sstem of inequalities for the given situation and graph the inequalities. + " 10, # 1. Write a sstem of inequalities for the following graph , * Solve each sstem of inequalities b graphing Ú Hours dog walking Nut Bread O O 8 (, ) Hours at car wash (, 1) (, 1) 8 O 8 (3, 3) Banana Bread O O O Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.