3.1 Open Sentences In Two Variables

Objective: To find solutions of open sentences in two variables

Chapter 3

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.

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)}

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

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

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

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

Homework

Page 104 – 105

#s 4, 6, 16, 18, 20, 22, 24, 26

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

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)}

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

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

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

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

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

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