SOLUTION: solve the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent. X+5Y=10 -2X-10Y=-20

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Question 121932: solve the following systems by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent.
X+5Y=10
-2X-10Y=-20
Found 2 solutions by jim_thompson5910, rapaljer:
Answer by jim_thompson5910(35256)   (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




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

Subtract from both sides

Divide both sides by 5.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute -10 to

Multiply



Reduce any fractions

Add to both sides


Combine the terms on the right side



Now combine the terms on the left side.
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 and we get

graph of


graph of (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
Answer by rapaljer(4671)   (Show Source): You can put this solution on YOUR website!
If you try to solve by the elimination method, multiply the first equation by 2 and add to the second equation:

2(X+5Y=10)
-2X-10Y=-20

2X+10Y=20
-2X-10Y=-20

The result is 0=0, which means that these equations are actually the SAME LINE. The equations are therefore said to be "Dependent."

R^2



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