SOLUTION: Please classify each system and determine the number of solutions. 1.) 7x+y=13 and 28x+4y=-2 2.) 2x-3y=-15 and 3y-2x=15 3.) 8y-24x=64 and 9y+45x=72 4.) 2x+2y=-10 and 4x+4y=-16

Algebra ->  College  -> Linear Algebra -> SOLUTION: Please classify each system and determine the number of solutions. 1.) 7x+y=13 and 28x+4y=-2 2.) 2x-3y=-15 and 3y-2x=15 3.) 8y-24x=64 and 9y+45x=72 4.) 2x+2y=-10 and 4x+4y=-16      Log On


   

Question 905669: Please classify each system and determine the number of solutions.
1.) 7x+y=13 and 28x+4y=-2
2.) 2x-3y=-15 and 3y-2x=15
3.) 8y-24x=64 and 9y+45x=72
4.) 2x+2y=-10 and 4x+4y=-16
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

1.)
7x%2By=13 and
28x%2B4y=-2
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


7x%2By=13

28x%2B4y=-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


7x%2By=13 Start with the given equation



1y=13-7x Subtract 7+x from both sides



1y=-7x%2B13 Rearrange the equation



y=%28-7x%2B13%29%2F%281%29 Divide both sides by 1



y=%28-7%2F1%29x%2B%2813%29%2F%281%29 Break up the fraction



y=-7x%2B13 Reduce



Now lets graph y=-7x%2B13 (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+-7x%2B13%29+ Graph of y=-7x%2B13




So let's solve for y on the second equation


28x%2B4y=-2 Start with the given equation



4y=-2-28x Subtract 28+x from both sides



4y=-28x-2 Rearrange the equation



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



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



y=-7x-1%2F2 Reduce





Now lets add the graph of y=-7x-1%2F2 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-7x%2B13%2C-7x-1%2F2%29+ Graph of y=-7x%2B13(red) and y=-7x-1%2F2(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.


2.)
2x-3y=-15 and
-2x%2B3y=15
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


2x-3y=-15

-2x%2B3y=15





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-3y=-15 Start with the given equation



-3y=-15-2x Subtract 2+x from both sides



-3y=-2x-15 Rearrange the equation



y=%28-2x-15%29%2F%28-3%29 Divide both sides by -3



y=%28-2%2F-3%29x%2B%28-15%29%2F%28-3%29 Break up the fraction



y=%282%2F3%29x%2B5 Reduce



Now lets graph y=%282%2F3%29x%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+%282%2F3%29x%2B5%29+ Graph of y=%282%2F3%29x%2B5




So let's solve for y on the second equation


-2x%2B3y=15 Start with the given equation



3y=15%2B2x Add 2+x to both sides



3y=%2B2x%2B15 Rearrange the equation



y=%28%2B2x%2B15%29%2F%283%29 Divide both sides by 3



y=%28%2B2%2F3%29x%2B%2815%29%2F%283%29 Break up the fraction



y=%282%2F3%29x%2B5 Reduce





Now lets add the graph of y=%282%2F3%29x%2B5 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%282%2F3%29x%2B5%2C%282%2F3%29x%2B5%29+ Graph of y=%282%2F3%29x%2B5(red) and y=%282%2F3%29x%2B5(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.


3.)
8y-24x=64 and
9y%2B45x=72
or
-24x%2B8y=64 and
45x%2B9y=72
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


-24x%2B8y=64

45x%2B9y=72





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


-24x%2B8y=64 Start with the given equation



8y=64%2B24x Add 24+x to both sides



8y=%2B24x%2B64 Rearrange the equation



y=%28%2B24x%2B64%29%2F%288%29 Divide both sides by 8



y=%28%2B24%2F8%29x%2B%2864%29%2F%288%29 Break up the fraction



y=3x%2B8 Reduce



Now lets graph y=3x%2B8 (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%2B8%29+ Graph of y=3x%2B8




So let's solve for y on the second equation


45x%2B9y=72 Start with the given equation



9y=72-45x Subtract 45+x from both sides



9y=-45x%2B72 Rearrange the equation



y=%28-45x%2B72%29%2F%289%29 Divide both sides by 9



y=%28-45%2F9%29x%2B%2872%29%2F%289%29 Break up the fraction



y=-5x%2B8 Reduce





Now lets add the graph of y=-5x%2B8 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+3x%2B8%2C-5x%2B8%29+ Graph of y=3x%2B8(red) and y=-5x%2B8(green)


From the graph, we can see that the two lines intersect at the point (0,8) (note: you might have to adjust the window to see the intersection)



4.)
2x%2B2y=-10 and
4x%2B4y=-16
Solved by pluggable solver: Solve the System of Equations by Graphing



Start with the given system of equations:


2x%2B2y=-10

4x%2B4y=-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


2x%2B2y=-10 Start with the given equation



2y=-10-2x Subtract 2+x from both sides



2y=-2x-10 Rearrange the equation



y=%28-2x-10%29%2F%282%29 Divide both sides by 2



y=%28-2%2F2%29x%2B%28-10%29%2F%282%29 Break up the fraction



y=-x-5 Reduce



Now lets graph y=-x-5 (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+-x-5%29+ Graph of y=-x-5




So let's solve for y on the second equation


4x%2B4y=-16 Start with the given equation



4y=-16-4x Subtract 4+x from both sides



4y=-4x-16 Rearrange the equation



y=%28-4x-16%29%2F%284%29 Divide both sides by 4



y=%28-4%2F4%29x%2B%28-16%29%2F%284%29 Break up the fraction



y=-x-4 Reduce





Now lets add the graph of y=-x-4 to our first plot to get:


+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x-5%2C-x-4%29+ Graph of y=-x-5(red) and y=-x-4(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.