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3.1 Open Sentences In Two Variables
Objective: To find solutions of open sentences in two variables
Chapter 3
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The x values are the inputs or the (domain), and the y values are the outputs or the (Range)
A solution of an open sentence is written as an ordered pair (x, y)
An Open sentence is an equation or inequality that contains one or more variables.
The following are some examples of open sentence:
3x = 1 + y3x = 1 + y x + y 5 >x + y 5 >
The set of all solutions to the open sentence is called the solution set.
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Solve y =4x – 6 if the domain of x is {-2, -1, 0}Example1:
If x = - 2 If x = - 2 then y = 4(–2) – 6 then y = 4(–2) – 6
= – 8 – 6 = – 8 – 6
= – 14 = – 14
Ordered pair (-2, -14) Ordered pair (-2, -14)
If x = - 1 If x = - 1 then y = 4(–1) – 6 then y = 4(–1) – 6
= – 4 – 6 = – 4 – 6
= – 10 = – 10
Ordered pair (-1, -10) Ordered pair (-1, -10)
If x = 0 If x = 0 then y = 4(0) – 6 then y = 4(0) – 6
= 0 – 6 = 0 – 6
= – 6 = – 6
Ordered pair (0, -6) Ordered pair (0, -6)
The Solution set is {(-2, -14), (-1, -10), (0, -6)}The Solution set is {(-2, -14), (-1, -10), (0, -6)}
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Complete each ordered pair to form a solution of the equation
Example2:
3x + 2y = 12 (0, __), (__, 0), (2, __)
If x = 0 If x = 0 then 3(0) + 2y = 12 then 3(0) + 2y = 12
2y = 122y = 12
y = 6 y = 6
Ordered pair (0, 6) 1st pair1st pair
If y = 0 If y = 0 then 3x + 2(0) = 12 then 3x + 2(0) = 12
3x = 123x = 12
x = 4 x = 4
Ordered pair (4, 0) 2nd pair2nd pair
If x = 2 If x = 2 Ordered pair (2, 3) 3rd pair3rd pair then 3(2) + 2y = 12
then 3(2) + 2y = 12 6 + 2y = 126 + 2y = 12
2y = 6 2y = 6
y = 3 y = 3
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Find the value of k so that the ordered pair satisfies the equation
Example3:
2x + y = k (2, 1)
Step1: Substitute the ordered pair in the equation
2(2) + (1) = k
Step2: solve for k
4 + 1 = k
5 = k
k = 5 k = 5
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Solve each equation if each variable represents a whole number
28 2x + y = 6 Whole numbers {0, 1, 2, 3, 4, 5, 6, 7, …….}
Rejected because -2 is not a whole
number
The Solution set is {(0, 6), (1, 4), (2, 2), (3, 0)}
The Solution set is {(0, 6), (1, 4), (2, 2), (3, 0)}
0
1
x 2x + y = 6 Ordered pair
2)0( + y = 6y = 6
)0 ,6(
2)1( + y = 6
2 + y = 6 )1 ,4(y = 4
22)2( + y = 6
4 + y = 6 )2 ,2(y = 2
32)3( + y = 6
6 + y = 6 )3 ,0(y = 0
42)4( + y = 6
8 + y = 6 )4 ,-2(y = -2
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Solve each equation if each variable represents a positive integer
34 2x + y > 6 Positive integers {1, 2, 3, 4, 5, 6, 7, 8, …….}
any number less than zero
is not a positive integer
The Solution set is {(1, 3), (1, 2), (1, 1), (2, 1)}
The Solution set is {(1, 3), (1, 2), (1, 1), (2, 1)}
1
x 2x + y < 6 Ordered pair
2)1( + y < 6
2 + y < 6 )1 ,3(y < 4
2
2)2( + y < 6
4 + y < 6 )2 ,1(y < 2
3
2)3( + y < 6
6 + y < 6
y < 0
y can be 3, 2 or 1
)1 ,2()1 ,1(
y can be 1
y can be none
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Homework
Page 104 – 105
#s 4, 6, 16, 18, 20, 22, 24, 26
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Solve each equation if the domain of x is {-1, 0, 2}
4
Written exercises page 104-105Written exercises page 104-105
-2x + y = -3
If x = - 1 If x = - 1 then -2x +y = -3 then -2x +y = -3 Ordered pair (-1, -5) Ordered pair (-1, -5)
The Solution set is {(-1, -5), (0, -3), (2, 1)}The Solution set is {(-1, -5), (0, -3), (2, 1)}
-2(-1) +y = -3 -2(-1) +y = -3 2 + y = -3 2 + y = -3 y = -5 y = -5
If x = 0 If x = 0 then -2x +y = -3 then -2x +y = -3 Ordered pair (0, -3) Ordered pair (0, -3)
-2(0) +y = -3 -2(0) +y = -3 0 + y = -3 0 + y = -3 y = -3 y = -3
If x = 2 If x = 2 then -2x +y = -3 then -2x +y = -3 Ordered pair (2, 1) Ordered pair (2, 1)
-2(2) +y = -3 -2(2) +y = -3 -4 + y = -3 -4 + y = -3 y = 1 y = 1
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6
Written exercises page 104-105Written exercises page 104-105
32
16 yx
Solve each equation if the domain of x is {-1, 0, 2}
If x = -1 If x = -1 Ordered pair (-1, -18) Ordered pair (-1, -18)
12(-1) – y = 6 12(-1) – y = 6 -12 – y = 6 -12 – y = 6 y = -18 y = -18
2bymultiply 612 yx
12x – y = 6 12x – y = 6
If x = 0 If x = 0 Ordered pair (0, -6) Ordered pair (0, -6)
12(0) – y = 6 12(0) – y = 6 0 – y = 6 0 – y = 6 y = -6 y = -6
12x – y = 6 12x – y = 6
If x = 2 If x = 2 Ordered pair (2, 18) Ordered pair (2, 18)
12(2) – y = 6 12(2) – y = 6 24 – y = 6 24 – y = 6 y = 18 y = 18
12x – y = 6 12x – y = 6
The Solution set is {(-1, -18), (0, -6), (2, 18)}The Solution set is {(-1, -18), (0, -6), (2, 18)}
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Complete each ordered pair to form a solution of the equation
16
Written exercises page 104-105Written exercises page 104-105
x + 6y = -9 (0, ___ ) ( ___, 0) (-3 , ___ )
Your TurnYour Turn
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Written exercises page 104-105Written exercises page 104-105
Complete each ordered pair to form a solution of the equation
18 3x + 5y = 3 (1, ___ ) ( ___, 7/5) (-2/3 , ___ )
Your TurnYour Turn
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Written exercises page 104-105Written exercises page 104-105
Complete each ordered pair to form a solution of the equation
20 (1, ___ ) ( ___, 6) (1/3 , ___ )23
1 yx
Your TurnYour Turn
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Find the value of k so that the ordered pair satisfies the equation
22
Written exercises page 104-105Written exercises page 104-105
3x - y = k (1 , -3)
Your TurnYour Turn
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Find the value of k so that the ordered pair satisfies the equation
24
Written exercises page 104-105Written exercises page 104-105
kx + 3y = 7 (-1 , 3)
Your TurnYour Turn
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Find the value of k so that the ordered pair satisfies the equation
26
Written exercises page 104-105Written exercises page 104-105
6x – ky = k (2 , 2)
Your TurnYour Turn