SOLUTION: Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent. 2x-y=4 2x-y=3

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Question 177967: Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
2x-y=4
2x-y=3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax-1%2Ay=4
2%2Ax-1%2Ay=3

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

-1%2Ay=4-2%2AxSubtract 2%2Ax from both sides

y=%284-2%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-4%2B2%2Ax Now we've fully isolated y

Since y equals -4%2B2%2Ax we can substitute the expression -4%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B-1%2Ahighlight%28%28-4%2B2%2Ax%29%29=3 Replace y with -4%2B2%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax-1%2A%28-4%29-1%282%29x=3 Distribute -1 to -4%2B2%2Ax

2%2Ax%2B4-2%2Ax=3 Multiply



2%2Ax%2B4-2%2Ax=3 Reduce any fractions

2%2Ax-2%2Ax=3-4 Subtract 4 from both sides


2%2Ax-2%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 2%2Ax-1%2Ay=4 (red) and 2%2Ax-1%2Ay=3 (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