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**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

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**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.

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**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)}

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**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

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**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

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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, …….} 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

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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, …….} 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

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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}**

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)}

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**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)}

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**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

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**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

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**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

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**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

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**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

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**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

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= (x + 6) (x + 2) 1. x2 +8x x2 +16x + 48 = (x + 12) (x + 4)

= (x + 6) (x + 2) 1. x2 +8x x2 +16x + 48 = (x + 12) (x + 4)

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