SOLUTION: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not. (a) 3X – Y = 3 (b) 2X – Y = 1

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not. (a) 3X – Y = 3 (b) 2X – Y = 1       Log On


   

Question 195945: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not.
(a) 3X – Y = 3 (b) 2X – Y = 1 (c) 2X – Y = 1
3X +Y = 15 2Y – 4X = 3 2Y – 4X = (-2)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the two problems to get you started

A)




Start with the given system of equations:
system%283x-y=3%2C3x%2By=15%29


Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:


%283x-y%29%2B%283x%2By%29=%283%29%2B%2815%29


%283x%2B3x%29%2B%28-y%2By%29=3%2B15 Group like terms.


6x%2B0y=18 Combine like terms.


6x=18 Simplify.


x=%2818%29%2F%286%29 Divide both sides by 6 to isolate x.


x=3 Reduce.


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3x-y=3 Now go back to the first equation.


3%283%29-y=3 Plug in x=3.


9-y=3 Multiply.


-y=3-9 Subtract 9 from both sides.


-y=-6 Combine like terms on the right side.


y=%28-6%29%2F%28-1%29 Divide both sides by -1 to isolate y.


y=6 Reduce.


So the solutions are x=3 and y=6.


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.




Graph of 3x-y=3 (red) and 3x%2By=15 (green)






B)

2y-4x=3 Start with the second equation.


-4x%2B2y=3 Rearrange the terms.




So we have the system of equations:
system%282x-y=1%2C-4x%2B2y=3%29


2%282x-y%29=2%281%29 Multiply the both sides of the first equation by 2.


4x-2y=2 Distribute and multiply.


So we have the new system of equations:
system%284x-2y=2%2C-4x%2B2y=3%29


Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:


%284x-2y%29%2B%28-4x%2B2y%29=%282%29%2B%283%29


%284x%2B-4x%29%2B%28-2y%2B2y%29=2%2B3 Group like terms.


0x%2B0y=5 Combine like terms.


0=5Simplify.


Since 0=5 is never true, this means that there are no solutions.

So the system is inconsistent.