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) 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

1%2Ax%2B5%2Ay=10
-2%2Ax-10%2Ay=-20

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=10-1%2AxSubtract 1%2Ax from both sides

y=%2810-1%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=2-%281%2F5%29%2Ax Now we've fully isolated y

Since y equals 2-%281%2F5%29%2Ax we can substitute the expression 2-%281%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-2%2Ax%2B-10%2Ahighlight%28%282-%281%2F5%29%2Ax%29%29=-20 Replace y with 2-%281%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

-2%2Ax-10%2A%282%29-10%28-1%2F5%29x=-20 Distribute -10 to 2-%281%2F5%29%2Ax

-2%2Ax-20%2B%2810%2F5%29%2Ax=-20 Multiply



-2%2Ax-20%2B2%2Ax=-20 Reduce any fractions

-2%2Ax%2B2%2Ax=-20%2B20Add 20 to both sides


-2%2Ax%2B2%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 1%2Ax%2B5%2Ay=10 and -2%2Ax-10%2Ay=-20 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%2810-1%2Ax%29%2F5%29+ graph of 1%2Ax%2B5%2Ay=10


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%28-20--2%2Ax%29%2F-10+%29+ graph of -2%2Ax-10%2Ay=-20 (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) About Me  (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