state whether the system has one solution, infinitely many solutions, or no solution. 3x + 7y = 2 6x + 14y = 4 4x + 5y = 3 6x + 9y = 9 6x + 11y = 12 12x + 22y = 10 0.3x + 0.7y = 1.2 6x + 14y = 24 Rule: Ax + By = C Dx + Ey = F Calculate AE and BD case 1. AE ¹ BD. Then there is exactly one solution. case 2. AE = BD. Then calculate AF and CD. If AF = CD then there are infinitely many solutions. If AF ¹ CD then there are no solutions. --------------------------------- --------------------------------- 3x + 7y = 2 6x + 14y = 4 A = 3, B = 7, C = 6, D = 6, E = 14, F = 4 Calculate AE and BD. AE = (3)(14) = 42 BD - (7)(6) = 42 This is case 2: case 2. AE = BD = 42. Then calculate AF and CD. AF = (3)(4) = 12 CD = (6)(2) = 12 AF = CD so there are infinitely many solutions. --------------------------------- 4x + 5y = 3 6x + 9y = 9 3x + 7y = 2 6x + 14y = 4 A = 4, B = 5, C = 3, D = 6, E = 9, F = 9 Calculate AE and BD. AE = (4)(9) = 36 BD - (5)(6) = 30 This is case 1, so there is exactly one solution. --------------------------------- 6x + 11y = 12 12x + 22y = 10 A = 6, B = 11, C = 12, D = 12, E = 22, F = 10 Calculate AE and BD. AE = (6)(22) = 132 BD - (11)(12) = 132 This is case 2: case 2. AE = BD = 132. Then calculate AF and CD. AF = (6)(10) = 60 CD = (12)(12) = 144 AF ¹ CD so there are no solutions. --------------------------------- 0.3x + 0.7y = 1.2 6x + 14y = 24 A = 0.3, B = 0.7, C = 1.2, D = 6, E = 14, F = 24 Calculate AE and BD. AE = (0.3)(14) = 4.2 BD - (0.7)(6) = 4.2 This is case 2: case 2. AE = BD = 4.2. Then calculate AF and CD. AF = (0.3)(24) = 7.2 CD = (1.2)(6) = 7.2 AF = CD so there are infinitely many solutions. Edwin