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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 y3 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 y2 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