SOLUTION: Which system of equations has no solution? 3x+6y=-6 3x-6y=-6 3x+6y=-6 9x+18y=-18 3x+6y=-6 3x+6y=-7 3x+6y=-6 5x-4y+-7

Algebra ->  Graphs -> SOLUTION: Which system of equations has no solution? 3x+6y=-6 3x-6y=-6 3x+6y=-6 9x+18y=-18 3x+6y=-6 3x+6y=-7 3x+6y=-6 5x-4y+-7      Log On


   

Question 105731: Which system of equations has no solution?
3x+6y=-6
3x-6y=-6
3x+6y=-6
9x+18y=-18
3x+6y=-6
3x+6y=-7
3x+6y=-6
5x-4y+-7
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

2.
3x%2B6y=-6
9x%2B18y=-18
there are no solutions, and the system is dependent+
the system is dependent that the equations have the same slope and the same intercept, and the lines are actually both the same line
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax%2B6%2Ay=-6
9%2Ax%2B18%2Ay=-18

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

Solve for y for the first equation

6%2Ay=-6-3%2AxSubtract 3%2Ax from both sides

y=%28-6-3%2Ax%29%2F6 Divide both sides by 6.


Which breaks down and reduces to



y=-1-%281%2F2%29%2Ax Now we've fully isolated y

Since y equals -1-%281%2F2%29%2Ax we can substitute the expression -1-%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


9%2Ax%2B18%2Ahighlight%28%28-1-%281%2F2%29%2Ax%29%29=-18 Replace y with -1-%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

9%2Ax%2B18%2A%28-1%29%2B18%28-1%2F2%29x=-18 Distribute 18 to -1-%281%2F2%29%2Ax

9%2Ax-18-%2818%2F2%29%2Ax=-18 Multiply



9%2Ax-18-9%2Ax=-18 Reduce any fractions

9%2Ax-9%2Ax=-18%2B18Add 18 to both sides


9%2Ax-9%2Ax=0 Combine the terms on the right side



0%2Ax=0 Now combine the terms on the left side.
0=0 Since this expression is true for any x, we have an identity.


So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph 3%2Ax%2B6%2Ay=-6 and 9%2Ax%2B18%2Ay=-18 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%28-6-3%2Ax%29%2F6%29+ graph of 3%2Ax%2B6%2Ay=-6


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%28-18-9%2Ax%29%2F18+%29+ graph of 9%2Ax%2B18%2Ay=-18 (hint: you may have to solve for y to graph these)

we can see that these two lines are the same. So this system is dependent



3.
3x%2B6y=-6
3x%2B6y=-7
there are no solutions, and the system is inconsistent

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax%2B6%2Ay=-6
3%2Ax%2B6%2Ay=-7

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

Solve for y for the first equation

6%2Ay=-6-3%2AxSubtract 3%2Ax from both sides

y=%28-6-3%2Ax%29%2F6 Divide both sides by 6.


Which breaks down and reduces to



y=-1-%281%2F2%29%2Ax Now we've fully isolated y

Since y equals -1-%281%2F2%29%2Ax we can substitute the expression -1-%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B6%2Ahighlight%28%28-1-%281%2F2%29%2Ax%29%29=-7 Replace y with -1-%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B6%2A%28-1%29%2B6%28-1%2F2%29x=-7 Distribute 6 to -1-%281%2F2%29%2Ax

3%2Ax-6-%286%2F2%29%2Ax=-7 Multiply



3%2Ax-6-3%2Ax=-7 Reduce any fractions

3%2Ax-3%2Ax=-7%2B6Add 6 to both sides


3%2Ax-3%2Ax=-1 Combine the terms on the right side



0%2Ax=-1 Now combine the terms on the left side.
0%2F1=-1%2F1 Since this expression is not true, we have an inconsistency.


So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.


graph of 3%2Ax%2B6%2Ay=-6 (red) and 3%2Ax%2B6%2Ay=-7 (green) (hint: you may have to solve for y to graph these)


and we can see that the two equations are parallel and will never intersect. So this system is inconsistent