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Warm-Up5 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 GraphingSolving Systems of Equations by GraphingSolving Systems of Equations by GraphingSolving Systems of Equations by Graphing
Objectives: •To find the solution of a system of equations by graphing
ActivityWith your partner:
Partner 1 - graph the equation 2x + 3y = 12Partner 2 - assist partner 1
Partner 2 - graph the equation x – 4y = -5Partner 1 - assist partner 2
How many points of intersection are there?
1
ActivityWith your partner:
Partner 1 - graph the equation x = 2y + 1Partner 2 - assist partner 1
Partner 2 - graph the equation 3x – 6y = 9Partner 1 - assist partner 2
How many points of intersection are there?
0
ActivityWith your partner:
Partner 1 - graph the equation 2x = 4 - yPartner 2 - assist partner 1
Partner 2 - graph the equation 6x + 3y = 12Partner 1 - assist partner 2
How many points of intersection are there?
an infinite number
Example 1Solve the following system by graphing.x + y = 2x = y
x + y = 2x y
21-3
015
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
x = y
x y
015
015
(1,1)
PracticeSolve by graphing.
1) x + 4y = -62x – 3y = -1
2) y + 2x = 52y – 5x = 10
Warm-Up4 minutes
1) y – 2x = 7y = 2x + 8
2) 3y – 2x = 64x – 6y = -12
Solve by graphing.
Example 1Determine whether (3,5) is a solution of the system.
y 4x 7
x y 8
y = 4x - 7 x + y = 8
5 =4( )3 - 7
5 = 12 - 75 = 5
3 + 5
= 8
8 = 8
(3,5) is a solution of the system
Example 2Determine whether (-2,1) is a solution of the system.
2x y 5
3x 2y 3
2x – y = -5 3x + 2y = 32(
)-2 -
1= -5-4 – 1 = -5
-5 = -5
3( )-2 + 2( )-6 + 2 = 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 – 2y = -4
PracticeSolve these systems by graphing.
1) x + 4y = -62x – 3y = -1
2) y + 2x = 52y – 5x = 10
Warm-Up1) Solve by graphing
y = 3x - 10
2) Solve for x where 5x + 3(2x – 1) = 5.
y = -2x + 10
The Substitution MethodThe Substitution MethodThe Substitution MethodThe Substitution MethodObjectives: •To solve a system of equations by substituting for a variable
Example 1Solve using substitution.y = 3x
2x + 4y = 28
2x + 4(3x) = 28
2x + 12x = 28 14x =
28x = 2
y = 3(2)
y = 6
(2,6)
y = 3x
PracticeSolve using substitution.
1) x + y = 5x = y
+ 1
2) a – b = 4b = 2 –
5a
Example 2Solve using substitution.
2x + y = 13
4x – 3y = 11y = -2x + 13
4x – 3(-2x + 13) = 11 4x + 6x – 39 =
11 10x – 39 = 11 10x =
50x = 52x + y =
132(5) + y = 1310 + y =
13 y = 3
(5,3)
PracticeSolve using substitution.
1) x = 2y + 82x + y = 26
2) 3x + 4y = 42y = 2x +
5
Example 3Solve 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 numberx + 2y =
13x = y + 4y + 4 + 2y = 13
3y + 4 = 13 3y = 9
y = 3x = y + 4x = 3 + 4x = 7
PracticeTranslate 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-UpSolve.
5 minutes
1)
y 5x 30
2x 2y 4
2)
6x y 18
3x 2y 18
8.3 The Addition Method8.3 The Addition Method8.3 The Addition Method8.3 The Addition MethodObjectives: •To solve a system of equations using the addition method
Example 1
x – y = 7 x + y = 3
2x + 0y = 102x = 10
x = 5
x + y = 35 + y = 3
y = -2
(5,-2)
Solve using the addition method.
Example 2
2x + 3y = 11-2x + 9y = 10x + 12y = 12 12y =
12y = 1
2x + 3y = 112x + 3(1) =
112x + 3 = 11 2x = 8
x = 4
(4,1)
Solve using the addition method.
Practice
1)
Solve using the addition method.
x y 5
2x y 4
2)
3x 3y 6
3x 3y 0
Example 3Solve using the addition method.3x – y =
8x + 2y = 5x + 2y = 5
write in standard form
3x – y = 8multiply as needed(-3)( )
( )(-3)
3x – y = 8-3x – 6y = -
15addition property
-7y = -7 y = 1
3x - (1) = 8 3x = 9
x = 3
(3,1)
Example 4
8x 2y 10
4x 15 3y
4x – 3y = 15
write in standard form
8x + 2y = -10 multiply as needed(-2)( ) ( )(-2)
8x + 2y = -10-8x + 6y = -30 addition property
8y = -40y = -
5
8x + 2(-5) = -10 8x - 10 = -
10 8x = 0x = 0(0,-5)
Solve using the addition method.
Practice
1)
Solve using the addition method.
5x 3y 17
5x 2y 3
2)
8x 11y 37
11y 7 2x
Warm-UpSolve.
5 minutes
1) y = 3x - 22x + 5y = 7
2) 5x – 2y = 42x + 4y = 16
8.4 Using Systems of Equations8.4 Using Systems of Equations8.4 Using Systems of Equations8.4 Using Systems of EquationsObjectives: •To solve problems using systems of equations
Example 1Translate 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-
busesLet r be the number of regular buses m
= 3r
m =
r + 60 3r = r +
60 2r = 60 r =
30
m = 3r m = 3(30) m =90 30 regular buses, 90 mini-
buses
PracticeTranslate 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 2Translate 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 BobLet f be the age of Fred
b =
f + 6 b
= 2f
f + 6 = 2f 6 = f
b = f + 6 b = (6) + 6b = 12
Bob is 12.Fred is 6.
Example 3Translate 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 now age 12 years ago
Fran
brother
f f - 12b b - 12
f =
b + 2 f – 12
= 2(b – 12) (b + 2) – 12 = 2(b –
12) b – 10 = 2b – 24 b = 2b – 14
b = 14f = b + 2f = 14 + 2f = 16Fran is 16; brother is 14
PracticeTranslate 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-Up5 minutesBeth 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 ProblemsDigit and Coin ProblemsDigit and Coin ProblemsDigit and Coin ProblemsObjectives: •To use systems of equations to solve digit and coin problems
Example 1The 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 digitLet y = the ones digit x + y =
1010y + x = 10x + y - 36 9y = 9x -
36y = x - 4x + y =
10x + (x – 4) = 10 2x - 4 =
10
2x - 4 = 10 2x =
14x = 7y = x - 4y = 7 - 4y = 373
PracticeThe 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 2A 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 + 0.10d = 3.95 d = 8 +
n0.05n + 0.10(8 + n) = 3.950.05n + 0.80 + 0.10n =
3.95 5n + 80 + 10n = 395 80 + 15n =
395 15n = 315n =
21
d = 8 + n d = 8 +
21d = 29
21 nickels29 dimes
PracticeRob has $2.85 in nickels and dimes. He has twelve more nickels than dimes. How many of each coin does he have?
Example 3There were 166 paid admissions to a game. The price was $2 for adults and $0.75 for children. The amount taken in was $293.25. How many adults and how many children attended?
Let a be the number of adults who attendedLet c be the number of children who attended
a + c = 1662a + 0.75c = 293.25a + c = 166a = 166 - c
2(166 – c) + 0.75c = 293.25332 – 2c + 0.75c = 293.25332 - 1.25c =
293.25- 1.25c = -38.75c =
31
a + c = 166a + 31 =
166 a = 135135 adults
31 children
PracticeThe attendance at a school concert was 578. Admission cost $2 for adults and $1.50 for children. The receipts totaled $985.00. How many adults and how many children attended the concert?
