SOLUTION: Solve and state if they are consistent, inconsistent, dependent and/or independent. 2x +3y = 1 6y = -4x + 2 2x+y = 5 x - y = 1 9x - 6y = 24 3x - 2y = 8 3x + 4y

Algebra ->  Inequalities -> SOLUTION: Solve and state if they are consistent, inconsistent, dependent and/or independent. 2x +3y = 1 6y = -4x + 2 2x+y = 5 x - y = 1 9x - 6y = 24 3x - 2y = 8 3x + 4y      Log On


   

Question 475866: Solve and state if they are consistent, inconsistent, dependent and/or independent.
2x +3y = 1
6y = -4x + 2
2x+y = 5
x - y = 1

9x - 6y = 24
3x - 2y = 8

3x + 4y = 12
6x + 8y = -16
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

2x+%2B3y+=+1
6y+=+-4x+%2B+2...write in standard form
2x+%2B3y+=+1
4x+%2B+6y+=2
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


2x%2B3y=1

4x%2B6y=2





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


2x%2B3y=1 Start with the given equation



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



3y=-2x%2B1 Rearrange the equation



y=%28-2x%2B1%29%2F%283%29 Divide both sides by 3



y=%28-2%2F3%29x%2B%281%29%2F%283%29 Break up the fraction



y=%28-2%2F3%29x%2B1%2F3 Reduce



Now lets graph y=%28-2%2F3%29x%2B1%2F3 (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+%28-2%2F3%29x%2B1%2F3%29+ Graph of y=%28-2%2F3%29x%2B1%2F3




So let's solve for y on the second equation


4x%2B6y=2 Start with the given equation



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



6y=-4x%2B2 Rearrange the equation



y=%28-4x%2B2%29%2F%286%29 Divide both sides by 6



y=%28-4%2F6%29x%2B%282%29%2F%286%29 Break up the fraction



y=%28-2%2F3%29x%2B1%2F3 Reduce





Now lets add the graph of y=%28-2%2F3%29x%2B1%2F3 to our first plot to get:


Graph of y=%28-2%2F3%29x%2B1%2F3(red) and y=%28-2%2F3%29x%2B1%2F3(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.




2x%2By+=+5
x+-+y+=+1

Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


2x%2By=5

1x-y=1





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


2x%2By=5 Start with the given equation



1y=5-2x Subtract 2+x from both sides



1y=-2x%2B5 Rearrange the equation



y=%28-2x%2B5%29%2F%281%29 Divide both sides by 1



y=%28-2%2F1%29x%2B%285%29%2F%281%29 Break up the fraction



y=-2x%2B5 Reduce



Now lets graph y=-2x%2B5 (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+-2x%2B5%29+ Graph of y=-2x%2B5




So let's solve for y on the second equation


1x-y=1 Start with the given equation



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



-y=-x%2B1 Rearrange the equation



y=%28-x%2B1%29%2F%28-1%29 Divide both sides by -1



y=%28-1%2F-1%29x%2B%281%29%2F%28-1%29 Break up the fraction



y=x-1 Reduce





Now lets add the graph of y=x-1 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-2x%2B5%2Cx-1%29+ Graph of y=-2x%2B5(red) and y=x-1(green)


From the graph, we can see that the two lines intersect at the point (2,1) (note: you might have to adjust the window to see the intersection)




9x+-+6y+=+24
3x+-+2y+=+8

Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


9x-6y=24

3x-2y=8





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


9x-6y=24 Start with the given equation



-6y=24-9x Subtract 9+x from both sides



-6y=-9x%2B24 Rearrange the equation



y=%28-9x%2B24%29%2F%28-6%29 Divide both sides by -6



y=%28-9%2F-6%29x%2B%2824%29%2F%28-6%29 Break up the fraction



y=%283%2F2%29x-4 Reduce



Now lets graph y=%283%2F2%29x-4 (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+%283%2F2%29x-4%29+ Graph of y=%283%2F2%29x-4




So let's solve for y on the second equation


3x-2y=8 Start with the given equation



-2y=8-3x Subtract 3+x from both sides



-2y=-3x%2B8 Rearrange the equation



y=%28-3x%2B8%29%2F%28-2%29 Divide both sides by -2



y=%28-3%2F-2%29x%2B%288%29%2F%28-2%29 Break up the fraction



y=%283%2F2%29x-4 Reduce





Now lets add the graph of y=%283%2F2%29x-4 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%283%2F2%29x-4%2C%283%2F2%29x-4%29+ Graph of y=%283%2F2%29x-4(red) and y=%283%2F2%29x-4(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.




3x+%2B+4y+=+12
6x+%2B+8y+=+-16

Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


3x%2B4y=12

6x%2B8y=-16





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%2B4y=12 Start with the given equation



4y=12-3x Subtract 3+x from both sides



4y=-3x%2B12 Rearrange the equation



y=%28-3x%2B12%29%2F%284%29 Divide both sides by 4



y=%28-3%2F4%29x%2B%2812%29%2F%284%29 Break up the fraction



y=%28-3%2F4%29x%2B3 Reduce



Now lets graph y=%28-3%2F4%29x%2B3 (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+%28-3%2F4%29x%2B3%29+ Graph of y=%28-3%2F4%29x%2B3




So let's solve for y on the second equation


6x%2B8y=-16 Start with the given equation



8y=-16-6x Subtract 6+x from both sides



8y=-6x-16 Rearrange the equation



y=%28-6x-16%29%2F%288%29 Divide both sides by 8



y=%28-6%2F8%29x%2B%28-16%29%2F%288%29 Break up the fraction



y=%28-3%2F4%29x-2 Reduce





Now lets add the graph of y=%28-3%2F4%29x-2 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28-3%2F4%29x%2B3%2C%28-3%2F4%29x-2%29+ Graph of y=%28-3%2F4%29x%2B3(red) and y=%28-3%2F4%29x-2(green)


From the graph, we can see that the two lines are parallel and will never intersect. So there are no solutions and the system is inconsistent.