3.2 Solving Systems of Equations Algebraically The Substitution Method: 1. Solve one of the equations for one of its variables. (coefficient of one on variable) 2. Substitute this expression into the other equation and solve for the other variable. 3. Substitute the value from #2 above in the revised first equation and solve. 4. Check the solution in each of the original equations. Solve the following problems by substitution. Ex. 1 4x − y = 9 x − 3 y = 16 The Elimination Method: 1. Arrange the equations with like terms in columns. Equations should look like standard form. 2. Obtain coefficients for x (or y) that have matching coefficients with opposite signs. If none match, multiply one or both to make a match. 3. Add so that the x column or y column is eliminated. 4. Solve for the remaining variable. 5. Substitute the value from #4 above into one of the original equations and solve for the other variable. 6. The solution should give an ordered pair, all reals, or empty set. Solve each problem below using elimination. Ex. 1 5a + 4b = 11 3a − 5b = −23 Ex. 2 5x − 3y = −3 2x + 6y = 0 heck 6 Solve the system using intersection. a graphing calculator. y ! x2 " 2 y ! "x Solve using any algebraic method: d 3 to find the second 1 x =1 4 y = x − 4Skill and Word Problem Practice. For more exercises, see2 Extra Intersection x + 2y = −8 3x − 4y = −24 y+ Sare (1, 4) and (4, 1). X=4 oblem Solving Y=1 aphing calculator. y ! x2 " 2 ! "x by graphing. Find the number of solutions for each system. mple Solve eachy system nd 2 753) 1. y ! x2 # 1 y!x#1 2. y ! x2 # 4 y ! 4x 3. y ! x2 " 5x " 4 y ! "2x Application: 4. y ! x2 # 2x # 4 5. y ! x2 # 2x # 5 6. y ! 3x # 4 A hair salon receives a shipment of 84 bottles of hair conditioner to use and exercises, see Extra y! x # 1Skill and Word Problem y ! "2xPractice. #1 y !sell "xto2 customers. The two types of conditioners received are type A, which is used for regular hair, and type B, which is used for frizzy hair. Type A costs $6.50 per bottle and type B Solve each using ple 3 costs $8.25 per system bottle. The hairelimination. salon’s invoice for the conditioner is $588. How many of each type of conditioner are in the shipment? 2 753) 7. y ! "x # 3 y!x #1 8. y ! x y!x#2 ing. Find the number of solutions for each system. 2 2. y ! x2 # 4 2 10. y ! x # 11 y ! 4x y ! "12x 3. y ! x2 " 5x " 4 11. y ! 5x " 20 y ! "2x y ! x2 " 5x # 5 9. y ! "x " 7 y ! x2 " 4x " 5 12. y ! x2 " x " 90 y ! x # 30 5. y !Solving x2 # Linear 2x # 5and Quadratic 6. yEquations ! 3x #Algebraically: 4 each system using substitution. 2 ple 4 y ! Solve "2x # 1 y ! "x 754) Ex. 13. y ! x2 " 2x " 6 mination. y ! 4x # 10 14. y ! 3x " 20 y ! "x2 # 34 15. y ! x2 # 7x # 100 y # 10x ! 30 17. " 3x7" y ! " 2 8. y ! 16. x2 "x2 " x # 19 !9.y y ! "x ! y5 y ! x #x2! y # 80 y ! x2 "2x 4x2 " 18. y ! 3x2 # 21x " 5 "10x # y ! "1 11. y ! 5x " 20 y ! x2 " 5x # 5 Practice: bstitution. 12. y ! x2 " x " 90 y ! x # 30 Lesson NY-6 Systems of Linear and Quadratic Equations 14. y ! 3x " 20 y ! "x2 # 34 15. y ! x2 # 7x # 100 y # 10x ! 30 17. 3x " y ! " 2 2x2 ! y 18. y ! 3x2 # 21x " 5 "10x # y ! "1 H.W. Pg. 164-165 #’s 4,6,10,16,20,22,28,30,32,36,55 Systems of Linear and Quadratic Equations NY 755 NY 755