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NAME 13-1 DATE Graphing Systems of Equations (Pages 550–553) 9.0 S T A N D A R D S A set of equations with the same variables forms a system of equations. A solution to a system of two or more equations is an ordered pair of numbers that satisfies all of the equations. One way to solve a system of equations is to carefully graph the equations on the same coordinate plane. The coordinates of the point at which the graphs intersect are the solution to the system. EXAMPLE Graph the system of equations to find the solution. y 2x 3 and y x 1 y The graphs appear to intersect at the point with coordinates (2, 1). Check this estimate by replacing x with 2 and y with 1 in each equation. Check: y 2x 3 yx1 1 2(2) 3 121 11✓ 11✓ The solution is (2, 1). (2, 1) x O y=x–1 y = 2x – 3 Try These Together Solve each system of equations by graphing. 1. y x 2 y 2x 1 2. y 2x 2 yx1 3. y 2x 1 y 3x 3 HINT: Be sure to check your solution by substituting the x- and y-values back into the two equations. PRACTICE Solve each system of equations by graphing. 4. y 9 x 5. 2x y 5 yx1 3x 3y 9 7. y 3 4x y 1 3 10. y x1 2 y 2x 6 B 4. 1 y x1 2 xy4 11. 2x 3y 6 12. 2x y 2 3x y 2 xy3 C B 8. 9. y x 2 C B A 7. 8. x y 2 C A 5. 6. 1 y4 x 3 B A 13. Standardized Test Practice Solve the system of equations. xy2 x y 10 A (3, 7) B (7, 3) C (4, 6) D (6, 4) Answers: 1–12. See Answer Key for graphs. 1. (3, 5) 2. (3, 4) 3. (2, 3) 4. (4, 5) 5. (8, 11) 6. (6, 2) 7. (1, 3) 8. (3, 1) 9. (2, 0) 10. (2, 2) 11. (0, 2) 12. (1, 4) 13. D 3. 6. y 8 x © Glencoe/McGraw-Hill 91 CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13-2 DATE Solutions of Systems of Equations (Pages 554–559) 9.0 S T A N D A R D S Graphs of systems of equations may be intersecting lines, parallel lines, or the same line. Systems of equations can be described by the number of solutions they have. A consistent system has at least 1 solution. An inconsistent system has no solution. y y (–1, 5) y x+y=2 y=x+2 x – y = –6 Graph 2x + y = 3 x O x O y=x–1 x O 3x + 3y = 6 Description of Graph intersecting lines same line parallel lines Number of Solutions 1 infinitely many 0 consistent and independent consistent and dependent inconsistent Type of System PRACTICE State whether each system is consistent and independent, consistent and dependent, or inconsistent. 1. 2. y 3. y y x O y = 2x – 1 y = –4x x O y = –4x – 6 O y = 3x + 2 x 6x + 2y = 4 4x – 2y = 2 Determine whether each system of equations has one solution, no solution, or infinitely many solutions by graphing. If the system has one solution, name it. 4. y x 4 1 x 2 B C B 8. 2 7. y 2x 4 3x y 5 C B A 7. x 1 y 3 C A 5. 6. xy5 B A 8. Standardized Test Practice What is the solution of the system y x 3 and y 2x 9? A Infinite solutions B no solution C (2, 5) 3. consistent and dependent 4–7. See Answer Key for graphs. 4. (4, 0) 4. 6. y 3x 6 © Glencoe/McGraw-Hill 92 D (5, 2) Answers: 1. inconsistent 2. consistent and independent 5. no solution 6. infinitely many 7. (1, 2) 8. C 3. y2 5. y x 6 CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13-3 DATE Substitution (Pages 560–565) 9.0, S T 25.0 A N D A R D S To solve a system of equations without graphing, you can use the substitution method shown in the example below. In general, if you solve a system of equations and the result is a true statement, such as 5 5, the system has infinitely many solutions; if the result is a false statement, such as 5 7, the system has no solution. EXAMPLE Use substitution to solve the system of equations x y 1 and 2x y 1. Step 1: Solve one of the equations for x or y. xy1 Solve the first equation for x since the x 1 y coefficient of x is 1. Step 3: Solve this equation. 2 2y y 1 y 3 or y 3 Step 2: Substitute this value into the other equation. 2x y 1 Use the second equation. 2(1 y) y 1 Substitute 1 y for x. 2 2y y 1 Distributive Property Step 4: Find the value of the other variable using substitution into either equation. xy1 Use the first equation. x31 Substitute 3 for y. x 2 Solve for x. The solution to the system is (2, 3). Check: Substitute 2 for x and 3 for y in each of the original equations and check for true statements. Try These Together Use substitution to solve each system of equations. 1. 3x y 19 x 2y 10 2. 2x y 7 8x y 3 3. y 2x 4 y 2x 2 4. y 5x 3 y 3x 3 HINT: If possible, choose to first solve an equation for a variable that has a coefficient of 1. PRACTICE Use substitution to solve each system of equations. 5. 5x 4 y y 3x 7 9. 2x y 4 x y 9 B 12. 5x y 98 2x 3y 5 C 13. Standardized Test Practice All CDs in the budget bin are priced the same. Packs of AA batteries are on sale. Keisha’s total bill (before tax) for 3 CDs and 1 pack of AA batteries was $39. Eduardo’s total for 2 CDs and 3 packs of batteries was $33. What was the price of a single CD? A $3 B $10 C $12 23 4 2 © Glencoe/McGraw-Hill 3 B A 93 3 B 8. 11. 3x y 28 x 3y 12 C B A 7. 10. 5x 2y 6 2x 3y 9 C A 5. 6. 8. 3y 4x 2 8x 6y 4 D $13 3 4. 7. 6x y 0 3x 4y 18 Answers: 1. (4, 7) 2. (1, 5) 3. no solution 4. , 5. , 6. 5, 7. , 4 8. infinitely many 9. (13, 22) 4 4 2 2 3 3 10. (0, 3) 11. (12, 8) 12. (23, 17) 13. C 3. 6. 3y x 1 2x 6 3y CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13-4 DATE Elimination Using Addition and Subtraction (Pages 566–571) 9.0 S T A N D A R D S In systems of equations where the coefficients of terms containing the same variable are opposites, the elimination method can be applied by adding the equations. If the coefficients of those terms are the same, the elimination method can be applied by subtracting the equations. EXAMPLES Solve each system of equations using elimination. A x 2y 13 and 3x 2y 15 B 3x 4y 5 and 3x y 5 Add the two equations, since the coefficients of the y-terms, 2 and 2, are opposites. Subtract the two equations, since the coefficients of the x-terms are the same. x 2y 13 () 3x 2y 15 4x 0 28 x7 Solve for x. 3x 4y 5 () 3x y 5 0 5y 10 y2 Solve for y. x 2y 13 Use the first equation. 7 2y 13 Substitute 7 for x. 2y 6 ⇒ y 3 3x y 5 Use the second equation. 3x 2 5 Substitute 2 for y. 3x 3 ⇒ x 1 The solution of the system is (7, 3). The solution of the system is (1, 2). Try These Together Use elimination to solve each system of equations 1. x y 3 3x y 1 2. 3x 4y 2 2x 4y 8 3. 2x 4y 8 y 7 2x PRACTICE Use elimination to solve each system of equations. 4. x 2y 3 x y 6 5. x y 2 xy8 6. 2y 3x 12 2y 6x 5 8. x 4y 16 2x 4y 18 9. 2x 4y 12 3x 4y 8 10. 8x y 4 8x 4y 8 7. 2x y 5 2x 3y 25 11. 2x 5y 7 2x 3y 1 12. Number Theory The sum of two numbers is 22. The greater number is two less than twice the other number. Write a system of equations to represent the problem. What are the numbers? B C C 8. C B A 13. Standardized Test Practice Solve the system. 2y 5x 1 3y 5x 14 A (3, 1) B (1, 3) 7 A 7. C (1, 3) © Glencoe/McGraw-Hill 19 B B 6. D (3, 1) 7 A 5. Answers: 1. (1, 2) 2. (6, 5) 3. (2, 3) 4. (3, 3) 5. (3, 5) 6. , 7. (5, 5) 8. 2, 9. (4, 1) 10. (1, 4) 3 2 2 4. 11. (1, 1) 12. x y 22 and x 2y 2; 14, 8 13. B 3. 94 CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13-5 DATE Elimination Using Multiplication (Pages 572–577) 9.0 S T A N D A R D S An extension of the elimination method is to multiply one or both of the equations in a system by some number so that adding or subtracting eliminates a variable. EXAMPLES Solve each system of equations using elimination. A x y 5 and 3x 2y 15 B 2x 9y 43 and 5x 2y 15 Multiply the first equation by 2 so that the coefficients of the y-terms in the system will be opposites. Then, add the equations and solve for x. 2(x y) 2(5) 3x 2y 15 ➞ Multiply the first equation by 5 and the second equation by 2 so that the coefficients of the x-terms in the system will be opposites. Then, add the equations and solve for y. 2x 2y 10 () 3x 2y 15 5x 0 25 x5 5(2x 9y) 5(43) 2(5x 2y) 2(15) xy5 Use the first equation. 5y5 Substitute 5 for x. y 0 ⇒ y 0 ➞ 2x 9y 43 2x 45 43 2x 2 ⇒ x 1 The solution to this system is (5, 0). 10x 45y 215 () 10x 4y 30 0 49y 245 y5 Use the first equation. Substitute 5 for y. The solution to the system is (1, 5). Try These Together Use elimination to solve each system of equations. 1. 2x y 4 3x 2y 6 2. 5x 2y 5 xy2 3. 4x 7y 6 6x 5y 20 4. x y 4 4x 2y 24 PRACTICE Use elimination to solve each system of equations. 5. 3x 4y 48 4x 3y 43 9. 2x 3y 0 3x y 7 6. 3x 8y 13 2x 5y 7 7. y 4x 11 3x 2y 7 1 10. 2x y 1 3 8. 2x 2y 16 3x y 4 11. 4x 2y 4 2x 3y 4 1 x y 8 4 12. Solve using elimination: 2x y 4 and x 2y 1. B C C B C 8. B A 13. Standardized Test Practice By which number could you multiply the first equation of the following system to solve the system by elimination? 4x 11y 32 and 12x 10y 55 A 3 or 3 B 10 or 10 C 11 or 11 D 12 or 12 7 A 7. 1 B 6. 5 A 5. Answers: 1. (2, 0) 2. (3, 5) 3. (5, 2) 4. (8, 4) 5. (4, 9) 6. (9, 5) 7. (3, 1) 8. (3, 5) 9. (3, 2) 10. , 18 2 4. 11. , 12. (3, 2) 13. A 4 2 3. © Glencoe/McGraw-Hill 95 CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13-6 DATE Solving Quadratic-Linear Systems of Equations (Pages 580–585) A quadratic-linear system of equations can have 0, 1, or 2 solutions. You can use graphing and substitution to solve quadratic-linear systems. y y y (–2, 2) x O O x O (1, –2) no solution one solution (1, –1) x two solutions EXAMPLE Solve. y x2 5 yx3 Substitute the values of x into one of the original equations to find values of y. Choose the easier equation to solve. Substitute x 3 for y in the first equation. x 3 x2 5 x x2 2 0 x2 x 2 0 (x 1)(x 2) 0x1 1 x or yx3 y 1 3 y 4 Add 3 to each side. Subtract x from each side. Factor. yx3 y23 y1 y The solutions are (1, 4) and (2, 1). The graph shows the solutions are correct. 0x2 2x x (2, –1) O (–1, –4) y = x2 – 5 y=x–3 PRACTICE Solve each system of equations by graphing. 1. y x2 3 y x 2. y x2 4 1 2 4. y x 1 2 3. y 5 y 2x2 5 yx2 y x 1 Use substitution to solve each system of equations. 5. x 2 y x2 1 B 4. C B C B A 7. 8. 8. y x 4 y x2 8 C A 5. 6. 7. y 2x 4 y x2 x 2 B A 9. Standardized Test Practice What is the solution of the system y 3x 3 and y x2 1? A (1, 0), (4, 15) B (0, 1), (15, 4) C no solution D (1, 0) Answers: 1–4. See Answer Key for graphs. 1. no solution 2. (2, 0), (3, 5) 3. (0, 5) 4. (2, 3), (0, 1) 5. (2, 3) 6. no solution 7. (2, 0), (3, 10) 8. (3, 1), (4, 8) 9. A 3. 6. y x2 5 yx1 © Glencoe/McGraw-Hill 96 CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13-7 DATE Graphing Systems of Inequalities (Pages 586–590) 9.0 S T A N D A R D S You can solve systems of inequalities by graphing. Recall that the graph of an inequality is a half-plane. The intersection of the two half-planes graphed in a system of inequalities represents the solution to the system. EXAMPLE Graph the system of inequalities to find the solution. x y 3 and y 3 x y y = –x + 3 Begin by solving each inequality for y. Then, graph each inequality. xy3 y x 3 and y3x yx3 x O y=x–3 The solution to the system includes the ordered pairs in the intersection of the graphs of each inequality. This region is shaded dark gray. Notice that the boundary lines y x 3 and y x 3 are included in the solution, since the inequalities contained and symbols. Try These Together Solve each system of inequalities by graphing. If the system does not have a solution, write no solution. 1. x 3 y5 2. x 4 y 1 3. y 3 x yx3 4. 2y x 6 3x y 4 HINT: Remember to graph inequalities with or with dashed lines because these lines are not included in the solution. PRACTICE Solve each system of inequalities by graphing. If the system does not have a solution, write no solution. 5. x 1 y 4 6. 2x y 4 3x y 6 7. y 2 x 2y 2 2x 8. x 4 y y 2 9. Money Carlita has some nickels and dimes in her pocket. She has at most 10 coins and they are worth at most 90 cents total. What combination of coins could she have? B C C A B 5. C B 6. A 7. 8. B A 10. Standardized Test Practice A dieter limits a snack to 90 Calories. Which is a possible snack combination of 20-Calorie apricots and 3-Calorie celery stalks? A 4 apricots 3 celery stalks B 3 apricots 10 celery stalks 10. D 4. © Glencoe/McGraw-Hill C 2 apricots 8 celery stalks D all of these Answers: 1–8. See Answer Key. 9. Sample answer: 5 nickels, 5 dimes 3. 97 CA Parent and Student Study Guide Algebra: Concepts and Applications NAME 13 DATE Chapter 13 Review 9.0 S T A N D A R D S Treasure Hunt Imagine that you and your parent are on a treasure hunt. The treasure hunt is taking place on a giant coordinate grid that is laid out on the floor of your school gym. You are competing with other parents and students for a grand prize. However, every parent/student team is looking for different treasures. The treasures for which you are searching are numbered stickers on the floor of this giant coordinate grid. Specifically, you are given a list of four items and a starting point. To locate the treasures, you must plot the intersection of the two graphs listed for each treasure. 2 Your starting point is the intersection of the graphs of y x 5 and 3 2y x 10. Find the coordinates of your starting point by graphing the two equations. Determine the coordinates for the location of each treasure by graphing each pair of equations given in the figure below. Then determine which treasure is closest to your starting point. The winner of the treasure hunt is the first parent/student team who turns in the treasure sticker that was closest to their starting point. 1 Gold: y 2x 8 and y x3 2 Silver: 3x 2y 6 and x 2y 10 1 1 Diamonds: x y 1 and y x7 4 4 Jewels: x 2y – 1 and y x 1 N Answers are located on page 121. © Glencoe/McGraw-Hill 98 CA Parent and Student Study Guide Algebra: Concepts and Applications