Download System of Equations

5 minutes
Warm-Up
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
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=2
x=y
x
y
0
2
8
1
1
6
4
5
-3
2
(1,1)
x=y
-8 -6
-4
-2
-2
-4
-6
-8
2
4
6
8
x
0
1
5
y
0
1
5
Practice
Solve by graphing.
1) x + 4y = -6
2x – 3y = -1
2) y + 2x = 5
2y – 5x = 10
4 minutes
Warm-Up
Solve by graphing.
1) y – 2x = 7
y = 2x + 8
2) 3y – 2x = 6
4x – 6y = -12
Example 1
Determine 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 - 7
5=5
3 + 5= 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  5

3x  2y  3
2x – y = -5
3x + 2y = 3
2(-2) - 1 = -5
-4 – 1 = -5
-5 = -5
3(-2) + 2( 1 ) = 3
-6 + 2 = 3
(-2,1) is not a solution of the system
Practice
Determine whether the given ordered pair is
a solution of the system.
1) (2,-3); x = 2y + 8
2x + y = 1
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 - 10
y = -2x + 10
2) Solve for x where 5x + 3(2x – 1) = 5.
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 = 3x
y = 3(2)
y=6
(2,6)
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 = 13
y = -2x + 13
4x – 3y = 11
4x – 3(-2x + 13) = 11
4x + 6x – 39 = 11
10x – 39 = 11
10x = 50
x=5
(5,3)
2x + y = 13
2(5) + y = 13
10 + y = 13
y=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 + 4 + 2y = 13
3y + 4 = 13
3y = 9
y=3
x=y+4
x=3+4
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.
5 minutes
Warm-Up
Solve.
1) y  5x  30
2x  2y  4
2) 6x  y  18
3x  2y  18
8.3 The Addition Method
Objectives:
•To solve a system of equations using the addition
method
Example 1
Solve using the addition method.
x–y=7
x+y=3
2x + 0y = 10
2x = 10
x=5
x+y=3
5+y=3
y = -2
(5,-2)
Example 2
Solve using the addition method.
2x + 3y = 11
-2x + 9y = 1
2x + 3y = 11
2x + 3(1) = 11
2x + 3 = 11
0x + 12y = 12
12y = 12
y=1
2x = 8
x=4
(4,1)
Practice
Solve using the addition method.
1) x  y  5
2x  y  4
2) 3x  3y  6
3x  3y  0
Example 3
Solve using the addition method.
3x – y = 8
x + 2y = 5
3x – y = 8
(-3)( x + 2y) =(5 )(-3)
3x – y = 8
-3x – 6y = -15
-7y = -7
y=1
write in standard form
multiply as needed
addition property
3x - (1) = 8
3x = 9
x=3
(3,1)
Example 4
Solve using the addition method.
8x  2y  10
4x  15  3y
8x + 2y = -10
(-2) (4x – 3y)= (15) (-2)
8x + 2y = -10
-8x + 6y = -30
8y = -40
y = -5
write in standard form
multiply as needed
addition property
8x + 2(-5) = -10
8x - 10 = -10
8x = 0
x=0
(0,-5)
Practice
Solve using the addition method.
1) 5x  3y  17
5x  2y  3
2) 8x  11y  37
11y  7  2x
5 minutes
Warm-Up
Solve.
1) y = 3x - 2
2x + 5y = 7
2) 5x – 2y = 4
2x + 4y = 16
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 + 60
m = 3r
3r = r + 60
m = 3(30)
2r = 60
m =90
r = 30
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=f+6
b = 2f
b = (6) + 6
f + 6 = 2f
6=f
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 now
Fran
brother
f
b
age 12 years ago
f - 12
b - 12
f= b+2
b = 14
f – 12 = 2(b – 12)
f=b+2
f = 14 + 2
(b + 2) – 12 = 2(b – 12)
f = 16
b – 10 = 2b – 24
b = 2b – 14
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?
5 minutes
Warm-Up
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
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
2x - 4 = 10
x + y = 10
10y + x = 10x + y - 36
2x = 14
x=7
9y = 9x - 36
y=x-4
y=x-4
x + y = 10
y=7-4
x + (x – 4) = 10
y=3
73
2x - 4 = 10
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 + 0.10d = 3.95
d=8+n
0.05n + 0.10(8 + n) = 3.95
d = 8 + 21
0.05n + 0.80 + 0.10n = 3.95
d = 29
5n + 80 + 10n = 395
21 nickels
80 + 15n = 395
29 dimes
15n = 315
n = 21
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 $293.25. 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 + 0.75c = 293.25
a + c = 166
a = 166 - c
2(166 – c) + 0.75c = 293.25
332 – 2c + 0.75c = 293.25
332 - 1.25c = 293.25
- 1.25c = -38.75
c = 31
a + c = 166
a + 31 = 166
a = 135
135 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 $985.00. How
many adults and how many children attended
the concert?