Need algebra help with 2 x 2 system or linear equations that are inconsistent or consistent dependent

label Algebra
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Nov 19th, 2015

Thank you for the opportunity to help you with your question!

Question 1

First, we need to add algebraically both equations. We add all what we have on the left side of each equation equal to the addition of all what we have on the right side of each equation. So we would have:

  x - 3y = - 3

- x + 3y = - 3

--------------------

x - x - 3y + 3y = - 3 - 3     --------------->   0 + 0 = - 6     ---------------->   0 = - 6 (false identity or equality).

As we can see all the x's and y's cancel out leaving a false equality. So when this happens then it means that the system has no solution.

Final answer: The system has no solution.


Question 2

  - x + 5y - 5  =   0 

    x - 5y      =  - 5

--------------------------

- x + x + 5y - 5y - 5 = 0 - 5    --------------->  0 + 0 - 5 = 0 - 5    ----------------> - 5 = - 5 (true equality or identity).

So again, as we can see all the x's and y's cancel out leaving a true equality (- 5 = - 5). So when this happens then it means that the system has infinitely many solutions. After this, we just need to choose the 1st r 2nd equation and we solve for y like this.

- x + 5y - 5 = 0    ------------> - x + x + 5y - 5 = 0 + x   ------------->  0 + 5y - 5 = x    -------------> 5y - 5 = x

5y - 5 = x  --------------> 5y - 5 + 5 = x + 5    ------------->  5y + 0 = x + 5  -------------> 5y = x + 5

5y = x + 5  ------------->  5y/5 = x/5 + 5/5    --------------> y = 1/5x + 1

Final answer:  The system has infinitely many solutions. 

They must satisfy the following equation: y = 1/5x + 1

Please let me know if you need any clarification. I'm always happy to answer your questions.
Nov 19th, 2015

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