SOLUTION: Identify the system of equations which has no solution. 5x + 2y = 4 2x – 2y = 10 –x = 3y + 1 x = 3y – 1 x – 2y = –6 2x = 4y – 12 x = 2y – 1 2x = 4y

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Question 418609: Identify the system of equations which has no solution.
5x + 2y = 4
2x – 2y = 10

–x = 3y + 1
x = 3y – 1

x – 2y = –6
2x = 4y – 12

x = 2y – 1
2x = 4y
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
Identify the system of equations which has no solution.
1.




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



Start with the given system of equations:










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


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce



Now lets graph (note: if you need help with graphing, check out this solver)



Graph of




So let's solve for y on the second equation


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce





Now lets add the graph of to our first plot to get:


Graph of (red) and (green)


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


2.








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



Start with the given system of equations:










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


Start with the given equation



Add to both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce



Now lets graph (note: if you need help with graphing, check out this solver)



Graph of




So let's solve for y on the second equation


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce





Now lets add the graph of to our first plot to get:


Graph of (red) and (green)


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



3.







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



Start with the given system of equations:










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


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce



Now lets graph (note: if you need help with graphing, check out this solver)



Graph of




So let's solve for y on the second equation


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce





Now lets add the graph of to our first plot to get:


Graph of (red) and (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.


4.









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



Start with the given system of equations:










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


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce



Now lets graph (note: if you need help with graphing, check out this solver)



Graph of




So let's solve for y on the second equation


Start with the given equation



Subtract from both sides



Rearrange the equation



Divide both sides by



Break up the fraction



Reduce





Now lets add the graph of to our first plot to get:


Graph of (red) and (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.



so, number 4. is your answer
Answer by richard1234(7193)   (Show Source): You can put this solution on YOUR website!
Instead of the extremely verbose yet tedious solution the other tutor posted using the pluggable solvers, we can determine which system has no solutions without solving each system or graphing.

First note that any two lines of different slope must intersect at exactly one point. Also, if two lines have the same slope, they are either parallel (no solution) or are the same line (infinitely many solutions). We start by finding the slopes of each line.

5x+2y = 4
2x-2y = 10

Rewriting each in y = mx + b form, we obtain:

y = (-5/2)x + 2
y = 2x - 10 Hence, this system has a solution.

Repeat the same for the other three systems:

y = (-1/3)x - 1/3
y = (1/3)x + 1/3, solution exists

y = (1/2)x + 3
y = (1/2)x + 3 These two lines are the same, so infinitely many solutions.

y = (1/2)x + 1/2
y = (1/2)x These lines are parallel and do not intersect. So, choice 4 has no solution.
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