SOLUTION: Find the intersection: 4x^2+20x+9y^2-18y-2=0 2x+3-2y=0

Question 747382: Find the intersection:
4x^2+20x+9y^2-18y-2=0
2x+3-2y=0
Answer by tinbar(133) (Show Source): You can put this solution on YOUR website!
The first equation will be hard to work with so we'll work with the second one. Usually when you are solving these kinds of questions, your main idea is to try and eliminate variables. What does this mean? Basically, it means, can you rewrite your equations in an equivalent manner to what was originally given but such that the new versions have lesser variables to solve in any or all the equations given.
So here's a small example:
If I say x+y = 5 and ask you to solve for x and y, you'll see there are an INFINITE amount of solutions. x=3,y=2 is one solution. x=1, y=4 is another and so on. No matter what though, we can never bring it down to some 'unique' solution because of the fact that there are TWO variables in this ONE equation.
Now let's say I claim x+y = 5 holds, but also that x - y = 1. Now how many solutions are there? It won't be infinitely many. Let's try the solutions we had from before. We can see that when x=3 and y = 2, x+ y = 5 is true AND that x - y =1 is also true. If we try x=1,y=4, while we satisfy x+y = 5, we do not satisfy x-y = 1. Similarly, we can find solutions that work for x-y = 1 but not for x+y=5. So the ONLY solution is x=3, y = 2.
Now how could I have solved it without guessing? Where does this "variable elimination" come into play?
Here's how we would have solved it:
x+y = 5
x-y = 1
From the first statement, x+y=5, we can rearrange and say it is equivalent to x = 5 - y. Now I haven't eliminated any variable yet since both x and y still show up in the equation, but we will use this new rearrangement to eliminate a variable in the second equation.
x-y = 1, and we also know x = 5 - y. Well, this "x" we speak of represents the same value in both equations, so whatever value we get for x in the second equation we will REPLACE it for instances of x in the first equation. So now in x- y = 1, we will replace all occurences of "x" with "5-y".
We now have (5-y) - y = 1. Look at what we just did! the x's have disappeared. Solving this is easy now.
5 - y - y = 1 ; 5 - 2y = 1 ; 5 - 1 = 2y; 4 = 2y ; y = 2. Yes! This is what we expected. Now how do we solve for x? Well, we know y, so let's go to both equations and replace y with this value.
So x+y = 5 becomes x+2 = 5
and x-y=1 becomes x-2 = 1
Now pick any one of them and solve x, both MUST give the same solution (Why? Think about this).
x+2 = 5; x = 5 - 2; x = 3. As expected
Now for your problem, repeat what I have done. Take your second equation and "isolate" a variable, that is re write it either as "x = ..." or "y = ... ". Once you have done this, replace either all x's or all y's (depending on your isolation) and solve out the first equation for the one variable that will be there, and then finally use either equation to solve your other variable.
Note that you may have more than one solution in certain systems. In the example I gave, there was only one, but this is not always the case.
For example, when the equation of a line cuts a parabola, it might have 0, 1 or 2 solutions.
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