SOLUTION: state weather 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.

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Question 121307This question is from textbook algebra2 explorations and applications
: state weather 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 This question is from textbook algebra2 explorations and applications

Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!

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
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