SOLUTION: I need help I am so confused..I need to solve each system by the substitution method. Determine whether the equations are independent, dependent, or incosistent..Thank you so much

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Question 124953: I need help I am so confused..I need to solve each system by the substitution method. Determine whether the equations are independent, dependent, or incosistent..Thank you so much
x-y=5 first system
2x=2y%2B14
Any help would be so greatly appreciated
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

x-y=5
2x=2y%2B14...divide both sides by 2 and you will have x=y%2B7...
then x+-+y+=7...


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


Lets start with the given system of linear equations

1%2Ax-1%2Ay=5
1%2Ax-1%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

-1%2Ay=5-1%2AxSubtract 1%2Ax from both sides

y=%285-1%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-5%2B1%2Ax Now we've fully isolated y

Since y equals -5%2B1%2Ax we can substitute the expression -5%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%28-5%2B1%2Ax%29%29=7 Replace y with -5%2B1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%28-5%29-1%281%29x=7 Distribute -1 to -5%2B1%2Ax

1%2Ax%2B5-1%2Ax=7 Multiply



1%2Ax%2B5-1%2Ax=7 Reduce any fractions

1%2Ax-1%2Ax=7-5 Subtract 5 from both sides


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



0%2Ax=2 Now combine the terms on the left side.
0%2F1=2%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 1%2Ax-1%2Ay=5 (red) and 1%2Ax-1%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