SOLUTION: how many solutions are for y + 3x = 2 and 2y - 4 = -6x ? one solution, no solution, infinite solutions, cannot be determined

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Question 868380: how many solutions are for y + 3x = 2 and 2y - 4 = -6x ?
one solution, no solution, infinite solutions, cannot be determined
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

if you are given this system:
y+%2B+3x+=+2 and
2y+-+4+=+-6x+
___________________________or
+3x%2By+=+2 and
6x%2B2y++=+4+
then,
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


3x%2By=2

6x%2B2y=4





In order to graph these equations, we need to solve for y for each equation.




So let's solve for y on the first equation


3x%2By=2 Start with the given equation



1y=2-3x Subtract 3+x from both sides



1y=-3x%2B2 Rearrange the equation



y=%28-3x%2B2%29%2F%281%29 Divide both sides by 1



y=%28-3%2F1%29x%2B%282%29%2F%281%29 Break up the fraction



y=-3x%2B2 Reduce



Now lets graph y=-3x%2B2 (note: if you need help with graphing, check out this solver)



+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-3x%2B2%29+ Graph of y=-3x%2B2




So let's solve for y on the second equation


6x%2B2y=4 Start with the given equation



2y=4-6x Subtract 6+x from both sides



2y=-6x%2B4 Rearrange the equation



y=%28-6x%2B4%29%2F%282%29 Divide both sides by 2



y=%28-6%2F2%29x%2B%284%29%2F%282%29 Break up the fraction



y=-3x%2B2 Reduce





Now lets add the graph of y=-3x%2B2 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-3x%2B2%2C-3x%2B2%29+ Graph of y=-3x%2B2(red) and y=-3x%2B2(green)


From the graph, we can see that the two lines are identical (one lies perfectly on top of the other) and intersect at all points of both lines. So there are an infinite number of solutions and the system is dependent.