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
