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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 + y 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 then y = 4(–2) – 6 = – 8 – 6 = – 14 Ordered pair (-2, -14) If x = - 1 then y = 4(–1) – 6 = – 4 – 6 = – 10 Ordered pair (-1, -10) If x = 0 then y = 4(0) – 6 = 0 – 6 = – 6 Ordered pair (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 then 3(0) + 2y = 12 2y = 12 y = 6 Ordered pair (0, 6) 1 st pair If y = 0 then 3x + 2(0) = 12 3x = 12 x = 4 Ordered pair (4, 0) 2 nd pair If x = 2 Ordered pair (2, 3) 3 rd pair then 3(2) + 2y = y = 12 2y = 6 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 = 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, …….} Rejected because -2 is not a whole number The Solution set is {(0, 6), (1, 4), (2, 2), (3, 0) } 0 1 x2x + y = 6 Ordered pair 2(0) + y = 6 y = 6 (0, 6) 2(1) + y = y = 6 (1, 4) y = 4 2 2(2) + y = y = 6 (2, 2) y = 2 3 2(3) + y = y = 6 (3, 0) y = 0 4 2(4) + y = 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) } 1 x2x + y < 6 Ordered pair 2(1) + y < y < 6 (1, 3) y < 4 2 2(2) + y < y < 6 (2, 1) y < 2 3 2(3) + y < 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 x + y = -3 If x = - 1 then -2x +y = -3 Ordered pair (-1, -5) The Solution set is {(-1, -5), (0, -3), (2, 1) } -2(-1) +y = y = -3 y = -5 If x = 0 then -2x +y = -3 Ordered pair (0, -3) -2(0) +y = y = -3 y = -3 If x = 2 then -2x +y = -3 Ordered pair (2, 1) -2(2) +y = y = -3 y = 1

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6 Written exercises page Solve each equation if the domain of x is {-1, 0, 2} If x = -1 Ordered pair (-1, -18) 12(-1) – y = – y = 6 y = x – y = 6 If x = 0 Ordered pair (0, -6) 12(0) – y = 6 0 – y = 6 y = -6 12x – y = 6 If x = 2 Ordered pair (2, 18) 12(2) – y = 6 24 – y = 6 y = 18 12x – y = 6 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 x + 6y = -9 (0, ___ ) ( ___, 0) (-3, ___ ) Your Turn

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

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

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

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

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

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