SOLUTION: what does an answer "no solution" mean graphically?

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Question 82478: what does an answer "no solution" mean graphically?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Say we had the system of equations

2x%2B5x=10
4x%2B10x=12

Lets solve the system of equations

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


Lets start with the given system of linear equations

2%2Ax%2B5%2Ay=10
4%2Ax%2B10%2Ay=12

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

5%2Ay=10-2%2AxSubtract 2%2Ax from both sides

y=%2810-2%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=2-%282%2F5%29%2Ax Now we've fully isolated y

Since y equals 2-%282%2F5%29%2Ax we can substitute the expression 2-%282%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B10%2Ahighlight%28%282-%282%2F5%29%2Ax%29%29=12 Replace y with 2-%282%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax%2B10%2A%282%29%2B10%28-2%2F5%29x=12 Distribute 10 to 2-%282%2F5%29%2Ax

4%2Ax%2B20-%2820%2F5%29%2Ax=12 Multiply



4%2Ax%2B20-4%2Ax=12 Reduce any fractions

4%2Ax-4%2Ax=12-20 Subtract 20 from both sides


4%2Ax-4%2Ax=-8 Combine the terms on the right side



0%2Ax=-8 Now combine the terms on the left side.
0%2F1=-8%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 2%2Ax%2B5%2Ay=10 (red) and 4%2Ax%2B10%2Ay=12 (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