SOLUTION: Solve the system of equations to find the point they intersect. 3x-6y=9 x-2y=3

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Question 111364: Solve the system of equations to find the point they intersect.
3x-6y=9
x-2y=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

3%2Ax-6%2Ay=9
1%2Ax-2%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

-6%2Ay=9-3%2AxSubtract 3%2Ax from both sides

y=%289-3%2Ax%29%2F-6 Divide both sides by -6.


Which breaks down and reduces to



y=-3%2F2%2B%281%2F2%29%2Ax Now we've fully isolated y

Since y equals -3%2F2%2B%281%2F2%29%2Ax we can substitute the expression -3%2F2%2B%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-2%2Ahighlight%28%28-3%2F2%2B%281%2F2%29%2Ax%29%29=3 Replace y with -3%2F2%2B%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-2%2A%28-3%2F2%29-2%281%2F2%29x=3 Distribute -2 to -3%2F2%2B%281%2F2%29%2Ax

1%2Ax%2B6%2F2-%282%2F2%29%2Ax=3 Multiply



1%2Ax%2B3-1%2Ax=3 Reduce any fractions

1%2Ax-1%2Ax=3-3 Subtract 3 from both sides


1%2Ax-1%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 3%2Ax-6%2Ay=9 and 1%2Ax-2%2Ay=3 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%289-3%2Ax%29%2F-6%29+ graph of 3%2Ax-6%2Ay=9


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%283-1%2Ax%29%2F-2+%29+ graph of 1%2Ax-2%2Ay=3 (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