Chapter 3
3.1 Open Sentences In Two Variables
Objective: To find solutions of open sentences in two variables
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.
Example1: Solve y =4x – 6 if the domain of x is {-2, -1, 0}
If x = - 2
then y = 4(–2) – 6
Ordered pair (-2, -14)
=–8–6
= – 14
If x = - 1
then y = 4(–1) – 6
Ordered pair (-1, -10)
=–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)}
Example2: Complete each ordered pair to form a solution of the
equation
3x + 2y = 12
1st pair
If x = 0
(0, __), (__, 0), (2, __)
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
6 + 2y = 12
2y = 6
y=3
Ordered pair (2, 3)
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
Solve each equation if each variable represents a whole number
28
2x + y = 6
x
2x + y = 6
0
2(0) + y = 6
y=6
Whole numbers {0, 1, 2, 3, 4, 5, 6, 7, …….}
Ordered pair
(0, 6)
2(1) + y = 6
1
2+y=6
y=4
(1, 4)
2(2) + y = 6
2
4+y=6
y=2
(2, 2)
2(3) + y = 6
3
6+y=6
y=0
(3, 0)
2(4) + y = 6
4
8+y=6
y = -2
(4, -2)
Rejected
because -2 is
not a whole
number
The Solution set is
{(0, 6), (1, 4), (2, 2), (3, 0)}
Solve each equation if each variable represents a positive integer
34
2x + y > 6
x
2x + y < 6
Positive integers {1, 2, 3, 4, 5, 6, 7, 8, …….}
Ordered pair
2(1) + y < 6
1
2+y<6
y<4
y can be 3, 2 or 1
(1, 3)
(1, 2)
(1, 1)
2(2) + y < 6
2
4+y<6
y<2
y can be 1
(2, 1)
any number
less than zero
is not a
positive integer
2(3) + y < 6
6+y<6
3
y<0
y can be none
The Solution set is
{(1, 3), (1, 2), (1, 1), (2, 1)}
Homework
Page 104 – 105
#s 4, 6, 16, 18, 20, 22, 24, 26
Written exercises page 104-105
Solve each equation if the domain of x is {-1, 0, 2}
4
-2x + y = -3
If x = - 1
then -2x +y = -3
-2(-1) +y = -3
If x = 0
then -2x +y = -3
-2(0) +y = -3
If x = 2
then -2x +y = -3
-2(2) +y = -3
Ordered pair (-1, -5)
2 + y = -3
y = -5
Ordered pair (0, -3)
0 + y = -3
y = -3
Ordered pair (2, 1)
-4 + y = -3
y=1
The Solution set is {(-1, -5), (0, -3), (2, 1)}
Written exercises page 104-105
Solve each equation if the domain of x is {-1, 0, 2}
6
6x 
1
y3
2
If x = -1
multiplyby 2
12x  y  6
12x – y = 6
12(-1) – y = 6
If x = 0
12x – y = 6
12(0) – y = 6
If x = 2
12x – y = 6
12(2) – y = 6
Ordered pair (-1, -18)
-12 – y = 6
y = -18
Ordered pair (0, -6)
0–y=6
y = -6
Ordered pair (2, 18)
24 – y = 6
y = 18
The Solution set is {(-1, -18), (0, -6), (2, 18)}
Written exercises page 104-105
Complete each ordered pair to form a solution of the equation
16
x + 6y = -9
(0, ___ ) ( ___, 0) (-3 , ___ )
Your Turn
Written 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 Turn
Written exercises page 104-105
Complete each ordered pair to form a solution of the equation
20
x
1
y2
3
(1, ___ ) ( ___, 6) (1/3 , ___ )
Your Turn
Written exercises page 104-105
Find the value of k so that the ordered pair satisfies the equation
22
3x - y = k
(1 , -3)
Your Turn
Written exercises page 104-105
Find the value of k so that the ordered pair satisfies the equation
24
kx + 3y = 7
(-1 , 3)
Your Turn
Written exercises page 104-105
Find the value of k so that the ordered pair satisfies the equation
26
6x – ky = k
(2 , 2)
Your Turn