SOLUTION: Solve system by the substitution method and indicate wheter each system is independent, inconsistent or dependent x-y=3 3x-2y=3

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Solve system by the substitution method and indicate wheter each system is independent, inconsistent or dependent x-y=3 3x-2y=3      Log On


   

Question 143588: Solve system by the substitution method and indicate wheter each system is independent, inconsistent or dependent
x-y=3
3x-2y=3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Start with the given system of equations:

system%28x-y=3%2C3x-2y=3%29



Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

x-y=3 Start with the first equation


-y=3-x Subtract x from both sides


-y=-x%2B3 Rearrange the equation


y=%28-x%2B3%29%2F%28-1%29 Divide both sides by -1


y=%28%28-1%29%2F%28-1%29%29x%2B%283%29%2F%28-1%29 Break up the fraction


y=x-3 Reduce



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

Since y=x-3, we can now replace each y in the second equation with x-3 to solve for x



3x-2highlight%28%28x-3%29%29=3 Plug in y=x-3 into the first equation. In other words, replace each y with x-3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



3x%2B%28-2%29%281%29x%2B%28-2%29%28-3%29=3 Distribute -2 to x-3


3x-2x%2B6=3 Multiply


x%2B6=3 Combine like terms on the left side


x=3-6Subtract 6 from both sides


x=-3 Combine like terms on the right side





-----------------First Answer------------------------------


So the first part of our answer is: x=-3









Since we know that x=-3 we can plug it into the equation y=x-3 (remember we previously solved for y in the first equation).



y=x-3 Start with the equation where y was previously isolated.


y=%28-3%29-3 Plug in x=-3


y=-3-3 Multiply


y=-6 Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is: y=-6









-----------------Summary------------------------------

So our answers are:

x=-3 and y=-6

which form the point




Since the system has a solution, this means that the system is independent and consistent.






Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of x-y=3 (red) and 3x-2y=3 (green) and the intersection of the lines (blue circle).