linear equations

Question

Several systems of equations are given below.

System 1
y = 6x – 1.5
y = –6x + 1.5

System 2
x + 3y = –6
2x + 6y = 3

System 3
2x –y = 5
6x – 3y = 15



Which system of equations is consistent-independent? How many solutions will the system of equations have? Explain your answers.
Which system of equations is consistent-dependent? How many solutions will the system of equations have? Explain your answers.
Which system of equations is inconsistent-independent? How many solutions will the system of equations have? Explain your answers.

Answer 1

Start with system 1:

y = 6x - 1.5

y = -6x + 1.5

 

By elimination y = 0...plugging that in:

0 = 6x - 1.5

1.5 = 6x

x = 1.5/6

x = .25

 

0 = -6x + 1.5

-1.5 = -6x

x = 1.5/6

x = .25

 

 

 

Check for truth:

0 = 6(.25) - 1.5

0 = 1.5 - 1.5

0 = 0 ; check

 

0 = -6(.25) + 1.5

0 = -1.5 + 1.5

0 = 0 ; check

 

System 1 is independent and consistent. It has only one solution and has at least one solution.

 

System 2: We'll use substitution here.

x + 3y = -6

2x + 6y = 3

 

3y = -x - 6

y = -(1/3)x - 2

2x + 6(-1/3)x -2 = 3

2x - 2x - 2 = 3

-2 = 3 ; nope

-2 ≠ 3

 

Doesn't work...so this system has no solutions. 

Therefore, the system is inconsistent and neither independent nor dependent since independent is defined as exactly one solution and dependent is defined as having an infinite number of solutions.

 

System 3:

2x - y = 5

6x -3y = 15

 

If you multiply the top equation by 3, you get the bottom equation which, when both are solved together, leads to the expression, 0 = 0:

Therefore the system has an infinite number of solutions.

Therefore, the system is dependent and consistent. Dependent because the system has infinite amount of solutions, and consistent because it has at least one solution.

 

Just to check, manipulate one of the equations to enable substitution:

2x - y = 5

-y = -2x + 5

y = 2x - 5

6x - 3(2x - 5) = 15

6x - 6x - 15 = 15

0 = 0: Infinite solutions

 

Answer 2

Hi Cody,

 

System 1 is consistent-independent because it has a single solution.  It is consistent because there the system has solution(s) and it is independent because there is only a single solution.

 

System 2 is inconsistent-independent or simply inconsistent because there are no solutions since the lines are parallel.

 

System 3 is consistent-dependent.  The lines have the same slope as in system 2 but in this case the intercept is identical and thus there are an infinite number of solutions because both equations define the identical line.  Since there are an infinite number of solutions, it is dependent.

Answer 3

System 1 is consistent independent

 

y = 6x-1.5 also y = -6x +1.5

 

Combining the 2 we get

 

6x-1.5= -6x +1.5

X = (1.5 + 1.5)/(12) =0.25

 

This has one solution

 

System 3 is consistent dependent

 

2x-y = 5

Multiplying by 3 we get

6x-3y= 15 

This is same as the second equation and hence it has infinite solutions and the system is consistent.

 

System 2 is inconsistent independent

X + 3y = -6

2X + 6Y = 3

 

Multiply equation 1 by 2

We get 2X + 6y =-12

Comparing to equation 2 we have -12 = 3

This is inconsistent and it has no solutions