Warm-Up 5 minutes 1) On the coordinate plane, graph two lines that will never intersect. 2) On the coordinate plane, graph two lines that intersect at one point. 3) On the coordinate plane, graph two lines that intersect at every point (an infinite number of points).

Solving Systems of Equations by Graphing Solving Systems of Equations by Graphing Objectives: To find the solution of a system of equations by graphing

Activity With your partner: Partner 1 - graph the equation 2x + 3y = 12 Partner 2 - assist partner 1 Partner 2 - graph the equation x – 4y = -5 Partner 1 - assist partner 2 How many points of intersection are there?1

Activity With your partner: Partner 1 - graph the equation x = 2y + 1 Partner 2 - assist partner 1 Partner 2 - graph the equation 3x – 6y = 9 Partner 1 - assist partner 2 How many points of intersection are there?0

Activity With your partner: Partner 1 - graph the equation 2x = 4 - y Partner 2 - assist partner 1 Partner 2 - graph the equation 6x + 3y = 12 Partner 1 - assist partner 2 How many points of intersection are there? an infinite number

Example 1 Solve the following system by graphing. x + y = 2 x = y x + y = 2 xy x = y xy (1,1)

Practice Solve by graphing. 1)x + 4y = -6 2x – 3y = -1 2)y + 2x = 5 2y – 5x = 10

Warm-Up 4 minutes 1) y – 2x = 7 y = 2x + 8 2) 3y – 2x = 6 4x – 6y = -12 Solve by graphing.

Example 1 Determine whether (3,5) is a solution of the system. y = 4x - 7x + y = 8 5 =4( ) = = = 8 8 = 8 (3,5) is a solution of the system

Example 2 Determine whether (-2,1) is a solution of the system. 2x – y = -53x + 2y = 3 2( ) = – 1 = = -5 3( ) ( ) = 3 1= 3 (-2,1) is not a solution of the system

Practice 1)(2,-3); x = 2y + 8 2x + y = 1 Determine whether the given ordered pair is a solution of the system. 2)(-3,4);2x = -y – 2 y = -4

Practice Solve these systems by graphing. 1)x + 4y = -6 2x – 3y = -1 2)y + 2x = 5 2y – 5x = 10

Warm-Up 1) Solve by graphing y = 3x ) Solve for x where 5x + 3(2x – 1) = 5. y = -2x + 10

The Substitution Method The Substitution Method Objectives: To solve a system of equations by substituting for a variable

Example 1 Solve using substitution. y = 3x 2x + 4y = 28 2x + 4(3x) = 28 2x + 12x = 28 14x = 28 x = 2 y = 3(2) y = 6 (2,6) y = 3x

Practice Solve using substitution. 1)x + y = 5 x = y + 1 2) a – b = 4 b = 2 – 5a

Example 2 Solve using substitution. 2x + y = 134x – 3y = 11 y = -2x + 134x – 3(-2x + 13) = 11 4x + 6x – 39 = 11 10x – 39 = 11 10x = 50 x = 5 2x + y = 13 2(5) + y = y = 13 y = 3 (5,3)

Practice Solve using substitution. 1)x = 2y + 8 2x + y = 26 2) 3x + 4y = 42 y = 2x + 5

Example 3 Solve using substitution. The sum of a number and twice another number is 13. The first number is 4 larger than the second number. What are the numbers? Let x = the first number Let y = the second number x + 2y = 13 x = y + 4 y y = 13 3y + 4 = 13 3y = 9 y = 3 x = y + 4 x = x = 7

Practice Translate to a system of equations and solve. 1) The sum of two numbers is 84. One number is three times the other. Find the numbers.

Warm-Up Solve. 5 minutes 1) 2)

8.3 The Addition Method 8.3 The Addition Method Objectives: To solve a system of equations using the addition method

Example 1 x – y = 7 x + y = 3 2x + 0y = 10 2x = 10 x = 5 x + y = y = 3 y = -2 (5,-2) Solve using the addition method.

Example 2 2x + 3y = 11 -2x + 9y = 1 0x + 12y = 12 12y = 12 y = 1 2x + 3y = 11 2x + 3(1) = 11 2x + 3 = 11 2x = 8 x = 4 (4,1) Solve using the addition method.

Practice 1) Solve using the addition method. 2)

Example 3 Solve using the addition method. 3x – y = 8 x + 2y = 5 write in standard form 3x – y = 8 multiply as needed (-3) ( ) (-3) 3x – y = 8 -3x – 6y = -15 addition property -7y = -7 y = 1 3x - (1) = 8 3x = 9 x = 3 (3,1)

Example 4 4x – 3y = 15 write in standard form 8x + 2y = -10 multiply as needed(-2) ( ) (-2) 8x + 2y = x + 6y = -30 addition property 8y = -40 y = -5 8x + 2(-5) = -10 8x - 10 = -10 8x = 0 x = 0 (0,-5) Solve using the addition method.

Practice 1) Solve using the addition method. 2)

Warm-Up Solve. 5 minutes 1) y = 3x - 2 2x + 5y = 7 2) 5x – 2y = 4 2x + 4y = 16

8.4 Using Systems of Equations 8.4 Using Systems of Equations Objectives: To solve problems using systems of equations

Example 1 Translate into a system of equations and solve. The Yellow Bus company owns three times as many mini-buses as regular buses. There are 60 more mini-buses than regular buses. How many of each does Yellow Bus own? Let m be the number of mini-buses Let r be the number of regular buses m =3r m =r r = r r = 60 r = 30 m = 3r m = 3(30) m =90 30 regular buses, 90 mini-buses

Practice Translate into a system of equations and solve. An automobile dealer sold 180 vans and trucks at a sale. He sold 40 more vans than trucks. How many of each did he sell?

Example 2 Translate into a system of equations and solve. Bob is 6 years older than Fred. Fred is half as old as Bob. How old are they? Let b be the age of Bob Let f be the age of Fred b =f + 6 b =2f f + 6 = 2f 6 = f b = f + 6 b = (6) + 6 b = 12 Bob is 12. Fred is 6.

Example 3 Translate into a system of equations and solve. Fran is two years older than her brother. Twelve years ago she was twice as old as he was. How old are they now? age nowage 12 years ago Fran brother ff - 12 b b - 12 f =b + 2 f – 12 =2(b – 12) (b + 2) – 12 = 2(b – 12) b – 10 = 2b – 24 b = 2b – 14 b = 14 f = b + 2 f = f = 16 Fran is 16; brother is 14

Practice Translate into a system of equations and solve. Wilma is 13 years older than Bev. In nine years, Wilma will be twice as old as Bev. How old is Bev?

Warm-Up 5 minutes Beth and Chris drove a total of 233 miles in 5.6 hours. Beth drove the first part of the trip and averaged 45 miles per hour. Chris drove the second part of the trip and averaged 35 miles per hour. For what length of the time did Beth drive?

Digit and Coin Problems Digit and Coin Problems Objectives: To use systems of equations to solve digit and coin problems

Example 1 The sum of the digits of a two-digit number is 10. If the digits are reversed, the new number is 36 less than the original number. Find the original number. Let x = the tens digit Let y = the ones digit x + y = 10 10y + x = 10x + y y = 9x - 36 y = x - 4 x + y = 10 x + (x – 4) = 10 2x - 4 = 10 2x = 14 x = 7 y = x - 4 y = y = 3 73

Practice The sum of the digits of a two-digit number is 5. If the digits are reversed, the new number is 27 more than the original number. Find the original number.

Example 2 A collection of nickels and dimes is worth $3.95. There are 8 more dimes than nickels. How many dimes and how many nickels are there? Let n be the number of nickels. Let d be the number of dimes. 0.05n d = 3.95 d = 8 + n 0.05n (8 + n) = n n = n n = n = n = 315 n = 21 d = 8 + n d = d = nickels 29 dimes

Practice Rob has $2.85 in nickels and dimes. He has twelve more nickels than dimes. How many of each coin does he have?

Example 3 There were 166 paid admissions to a game. The price was $2 for adults and $0.75 for children. The amount taken in was $ How many adults and how many children attended? Let a be the number of adults who attended Let c be the number of children who attended a + c = 166 2a c = a + c = 166 a = c 2(166 – c) c = – 2c c = c = c = c = 31 a + c = 166 a + 31 = 166 a = adults 31 children

Practice The attendance at a school concert was 578. Admission cost $2 for adults and $1.50 for children. The receipts totaled $ How many adults and how many children attended the concert?