1. 3.1 Open Sentences In Two Variables
Chapter 3
3.1 Open Sentences In Two Variables
Objective: To find solutions of open sentences in two variables

2. An Open sentence is an equation or inequality that contains one or
more variables.
The following are some examples of open sentence: 
3x = 1 + y
x + y 5 >
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)
The set of all solutions to the open sentence is called the solution set.

3. Solve y =4x – 6 if the domain of x is {-2, -1, 0}
Example1:
If x = - 2
then y = 4(–2) – 6
Ordered pair (-2, -14)
= – 8 – 6
= – 14
then y = 4(–1) – 6
Ordered pair (-1, -10)
If x = - 1
= – 4 – 6
= – 10
If x = 0
then y = 4(0) – 6
Ordered pair (0, -6)
= 0 – 6
= – 6
The Solution set is {(-2, -14), (-1, -10), (0, -6)}

4. Complete each ordered pair to form a solution of the equation
Example2:
Complete each ordered pair to form a solution of the equation
3x + 2y = 12
(0, __), (__, 0), (2, __)
1st pair
If x = 0
then 3(0) + 2y = 12
Ordered pair (0, 6)
2y = 12
y = 6
2nd pair
If y = 0
then 3x + 2(0) = 12
Ordered pair (4, 0)
3x = 12
x = 4
3rd pair
If x = 2
then 3(2) + 2y = 12
Ordered pair (2, 3)
6 + 2y = 12
2y = 6
y = 3

5. Find the value of k so that the ordered pair satisfies the equation
Example3:
Find the value of k so that the ordered pair satisfies the equation
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

6. Solve each equation if each variable represents a whole number28
2x + y = 6
Whole numbers {0, 1, 2, 3, 4, 5, 6, 7, …….} x
2x + y = 6
Ordered pair
2(0) + y = 6
(0, 6)
y = 6
2(1) + y = 6 1
(1, 4)
2 + y = 6
y = 4
Rejected because -2 is not a whole number
2(2) + y = 6 2
(2, 2)
4 + y = 6
y = 2
2(3) + y = 6 3
(3, 0)
6 + y = 6
y = 0
2(4) + y = 6
The Solution set is {(0, 6), (1, 4), (2, 2), (3, 0)} 4
(4, -2)
8 + y = 6
y = -2

7. Solve each equation if each variable represents a positive integer34
2x + y > 6
Positive integers {1, 2, 3, 4, 5, 6, 7, 8, …….} x
2x + y < 6
Ordered pair
2(1) + y < 6 1
(1, 3)
2 + y < 6
(1, 2)
y < 4
(1, 1)
y can be 3, 2 or 1
any number less than zero is not a positive integer
2(2) + y < 6
4 + y < 6 2
(2, 1)
y < 2
y can be 1
2(3) + y < 6
6 + y < 6
The Solution set is {(1, 3), (1, 2), (1, 1), (2, 1)} 3
y < 0
y can be none

8. Homework
Page 104 – 105
#s 4, 6, 16, 18, 20, 22, 24, 26

9. Solve each equation if the domain of x is {-1, 0, 2}
Written exercises page
Solve each equation if the domain of x is {-1, 0, 2} 4
-2x + y = -3
If x = - 1
then -2x +y = -3
Ordered pair (-1, -5)
-2(-1) +y = -3
2 + y = -3
y = -5
If x = 0
then -2x +y = -3
Ordered pair (0, -3)
-2(0) +y = -3
0 + y = -3
y = -3
If x = 2
then -2x +y = -3
Ordered pair (2, 1)
-2(2) +y = -3
-4 + y = -3
y = 1
The Solution set is {(-1, -5), (0, -3), (2, 1)}

10. Solve each equation if the domain of x is {-1, 0, 2}
Written exercises page
Solve each equation if the domain of x is {-1, 0, 2} 6
If x = -1
Ordered pair (-1, -18)
12x – y = 6
12(-1) – y = 6
-12 – y = 6
y = -18
If x = 0
Ordered pair (0, -6)
12x – y = 6
12(0) – y = 6
0 – y = 6
y = -6
If x = 2
Ordered pair (2, 18)
12x – y = 6
12(2) – y = 6
24 – y = 6
y = 18
The Solution set is {(-1, -18), (0, -6), (2, 18)}

11. Complete each ordered pair to form a solution of the equation
Written exercises page
Complete each ordered pair to form a solution of the equation
(0, ___ ) ( ___, 0) (-3 , ___ ) 16
x + 6y = -9
Your Turn

12. Complete each ordered pair to form a solution of the equation
Written exercises page
Complete each ordered pair to form a solution of the equation
(1, ___ ) ( ___, 7/5) (-2/3 , ___ ) 18
3x + 5y = 3
Your Turn

13. Complete each ordered pair to form a solution of the equation
Written exercises page
Complete each ordered pair to form a solution of the equation
(1, ___ ) ( ___, 6) (1/3 , ___ ) 20
Your Turn

14. Find the value of k so that the ordered pair satisfies the equation
Written exercises page
Find the value of k so that the ordered pair satisfies the equation
(1 , -3) 22
3x - y = k
Your Turn

15. Find the value of k so that the ordered pair satisfies the equation
Written exercises page
Find the value of k so that the ordered pair satisfies the equation
(-1 , 3) 24
kx + 3y = 7
Your Turn

16. Find the value of k so that the ordered pair satisfies the equation
Written exercises page
Find the value of k so that the ordered pair satisfies the equation
(2 , 2) 26
6x – ky = k
Your Turn