Chapters 3 & 4 Algebra 2 Honors 1 Systems of Equations How many solutions do each have, and what do those solutions look like graphically? 3x + 1 =4 3x + 1 > 4 3x + 2y = 4 3x + 2y > 4 2x - y = 4 x + 4y = -7 A system of equations : Is (0, 2) a solution to the above system? Is ( 0, -4)? Is ( 1, -2)? How many solutions does this system have?
Chapters 3 & 4 Algebra 2 Honors 2 Independent systems: Dependent systems: Inconsistent systems: 4 methods for solving a system of equations: 2x y = 5 y = -x + 1 Solving by Substitution
Chapters 3 & 4 Algebra 2 Honors 3 3x 2y = -11 4x + y = 9 x 2y = 9 1/2x 4y = 2 Be on the lookout for inconsistent and dependent solutions. When solving by substitution: Inconsistnet systems : Dependent systems: 3x 12y = -24 -x + 4y = 8 Page 120 13, 15, 17 27, 29, 35
Chapters 3 & 4 Algebra 2 Honors 4 Solving by the Addition (Elimination) Method x + y = 1 x + 2y = 4 2x y = 5 2x + y = -1 2x 3y = 6 3x + 6y = 7 3x + 8y = -9 2x + 4y = 5 Page 120 19, 21, 23, 29, 31, 35
Chapters 3 & 4 Algebra 2 Honors 5 3x + 4 = y x 1 = y 1 3 3 x y 4 3 2 5 x y 12 y = 2x + 1 2x y = 1 3 x x 2 y 5 x 2 x 1 = y Page 120 29, 31, 35
Chapters 3 & 4 Algebra 2 Honors 6 Solving by Graphing 2x 5 x 1 y 5x 2 x 1 y= 4 3x y = -5 3x 2y = 4 y = 2x - 1 y = 3 y = -3x + 2 3x 2y = 6 ans ans.
Chapters 3 & 4 Algebra 2 Honors 7 y = x + 1 2x + 3y = 6 x 5y = 3 y = -2/3x + 1 ans. ans. Be on the lookout for inconsistent and dependent solutions. When solving by graphing: Inconsistnet systems : Dependent systems: Page 113 13, 17, 23, 25, 28, 68, 69
Chapters 3 & 4 Algebra 2 Honors 8 Matrix: Matrices Used to solve many different types of math problems. Sometimes thought of as the coefficient of terms in equations. Noted by Captial letters. Size is given in rows X columns A = 3 1 2 0 7 2 Square matrix Determinant of a matrix: Find the determinant of B = 3 4 1 2 3 1 6 2
Chapters 3 & 4 Algebra 2 Honors 9 2 1 2 1 5 0 5 0 3 5 Page 186 15, 17, 19, 21, 23, 25 Solving Systems using Matrices Cramer s Rule 2x 3y = 8 5x + 6y = 11 1. Create and find determinant of coeff matrix 2. Create and find determinant of x matrix
Chapters 3 & 4 Algebra 2 Honors 10 3. Create and find determinant of y matrix 4. Find x by dividing: 5. Find y by dividing: Solve using Cramer s Rule 6x 6y = 5 2x 5y = 26 2x 10y = -1 5x + 3y = 3 Page 192 13, 15, 23
Chapters 3 & 4 Algebra 2 Honors 11 Solve using Cramer s Rule 3x 2y = 6 4x 8y = 36 6x 4y = 4 3x - 6y = 27 When solving a system using Cramer s Rule we find inconsistent systems, no solutions, when When solving a system using Cramer s Rule we find dependent systems, solutions, when Remember to be on the lookout for inconsistent and dependent systems. Pg 120 17, 25 use Cramer s
Chapters 3 & 4 Algebra 2 Honors 12 Solve for x and y ax + by = c dx + ey = f Word Problems!!!!!!!!!!!! All problems must be solved by solving equations. All variables used must be defined in words. All problems must be answered in sentence form. 1. The larger of two numbers is 1 less than three times the smaller. The difference between the two numbers is 9. Find the numbers. 2. If 14 is added to a number and that sum is multiplied by 2, the result is equal to 8 times the number decreased by 14. Find the number. 3. Twice the smaller of two numbers is one-half the larger number. The larger is 10 more than 3 times the smaller. 4. The sum of 1/3 of the smaller of two numbers and 2/3 of the larger is 34. Also, ½ of the smaller number is equal to ¼ of the larger number. Find the numbers.
Chapters 3 & 4 Algebra 2 Honors 13 5. Find three consecutive integers such that first increased by twice the second exceeds the third by 24. 6. Find three consecutive odd integers such that product of the second and third exceeds the square of the first by 50. 7. Find three consecutive even integers such that the sum of the smallest and twice the second is 20 more than the third. 8. The sum of there consecutive integers is 57. Find the integers. 9 A cup contains dimes and nickels. There are 45 coins. The value of the coins is $3.50. How many dimes are there? 10. Wanda has 25 coins worth $3.20. There are nickels, dimes and quarters. The number of dimes exceeds the number of nickels by 4. How many of each coin does she have? 11. There is a stack of five, ten and twenty-dollar bills. The value of the bills is $485. There are twice as many ten-dollar bills as there are twenty dollar bills and 7 more five-dollar bills than ten-dollar bills. How many five-dollar bills are there? 12. There are 3 fewer quarters than nickels and 4 times as many dimes as nickels. There is $4.15. How many of each coin is there? 13. Walter is 3 times as old as Martin. Ten years from now, Walter will be twice as old as Martin will be then. How old is each now? 14. The sum of Carl s age and Ellen s age is 40 years. Carl s age 10 years from now will be 1 year less than 4 times Ellen s age 6 years ago. What are their present ages? 15. Ray is 20 years older than Bill. Five years ago, Ray was 5 times as old as Bill was then. How old is Ray right now? 16. Sidney is 30 years old and Edward is 15 years old. In how many years will Sidney be 1.5 times as old as Edward will be then?
Chapters 3 & 4 Algebra 2 Honors 14 17. The ratio of Paul s age to Joy s age is 3 : 5. The sum of their ages is 48 years. How old are Paul and Joy? 18. A 180 degree angle is divided into three angles. Two angles are in the ratio of 2 : 3. The third angle is equal to the sum of the degrees of the other two angles. Find the degree measure of each of the three angles. 19. There are 54 marbles. The marbles are red, white and blue. The ratio of the number of each color, red, white and blue respectively is 3 : 7 : 8. How many red marbles are there? 20. A will states that Gail, you and Pepe are to split the inheritance in the respective ratios of 1 : 3 : 7. If there is $3,300 to split, how much will you receive? The above problems we skipped 2x y 3 3x + 2y > 8 Solving Systems of Linear Inequalities
Chapters 3 & 4 Algebra 2 Honors 15 y 1/2x 3 y x 1 y < -2 y 1/2x + 2 y 1/2x + 1 y x 1 y > x 1 y < 3
Chapters 3 & 4 Algebra 2 Honors 16 Be on the lookout for inconsistent systems. 2x + 3y > 9 y < -2/3x + 1 Page 126 13, 15, 17, 19, 23
Chapters 3 & 4 Algebra 2 Honors 17 Linear Programming Linear programming Find the maximum and minimum value, if they exist, for the function f(x, y) = 2x + 3y given the following constarints: x 2 y -1 y 2x + 1 Find the maximum and minimum value, if they exist, for the function f(x, y) = 5x y given the following constraints: 2x + y 3 3y x 9 2x + y 10 Page 132 15, 19, 21
Chapters 3 & 4 Algebra 2 Honors 18 Solve x + 2y z = 1 x 2y + z = 6 2x y + z = 6 3x + y 2z = 2 x + 3y z = 2 2x 3y + 2z = -7 Page 142 13, 15, 25
Chapters 3 & 4 Algebra 2 Honors 19 2 1 4 2 1 3 5 0 1 0 1 2 3 5 3 4 0 2 Det with calculator Page 186 27, 31
Chapters 3 & 4 Algebra 2 Honors 20 Solve using Cramer s Rule: 3x 2y + z = 2 2x + 3y + 2z = -6 3x y + z = 0 x 2y + z = 6 3x + y 2z = 2 2x 3y + 2z = -7 Page 192 27, 29, 36, 37
Chapters 3 & 4 Algebra 2 Honors 21 Find the area of the triangle with vertices (-1, 5), (-1, 1) and (4, 1). Find the area of the trapezoid with vertices (2, 3) (2, -3) (5, -3) and (5, 1). Page 186 39, 40, 41