MEDAL!? Find the intersection points, if any, for each system of equations: (x^2/4)+y^2=1
y=x+1

Question

MEDAL!? Find the intersection points, if any, for each system of equations: (x^2/4)+y^2=1
                     y=x+1

marigirl · Answer

The first equation is an ellipse. so both the equations can be graphed

marigirl · Answer

attachment

marigirl · Answer

So then the graphs look like this

marigirl · Answer

then we can find the points of intersections:
(0,1) and (-1.6,-0.6)

marigirl · Answer

that was done graphically.. Does your answer need to be done algebraically? Let me know :)

anonymous · Answer

Is that what I would write as my answer then? If it is not too much trouble could you write about the process?

marigirl · Answer

What I did above was to graph it using a graphics calculator and find the points of intersection through that. If you were to solve it algebraically we can use the substitution method:
(x^2/4)+y^2=1 and y=x+1 using substition
$$\frac{ x^2 }{ 4 }+(x+1)^2=1$$ Then we solve for x and the two x solutions that you will obtain will be x=0 and x=-1.6. Then you take those two values and plug it into any of the two original equations to obtain the coordinates (0,1) and (-1.6,-0.6)

So the line y=x+1 crosses the ellipse at two places.

anonymous · Answer

Oh ok! What about x^2+y^2-16y+39=0
                             y^2-x^2-9=0

marigirl · Answer

ellipse and hyperbola..

anonymous · Answer

ANy intersection points?

marigirl · Answer

Make y the subject of the second equation and then substitute into the first..give me some time for the intersections lol

marigirl · Answer

THere are points that intersect  but it is not showing up on my calc :(

anonymous · Answer

try desmos.com/calculator

marigirl · Answer

Ill tell you what I have so far to graph this 
The ellipse: $$x=\sqrt{-y^2+16y-39}$$

marigirl · Answer

$$x=-\sqrt{-y^2+16y-39}$$

marigirl · Answer

The Hyperbola
$$y=\sqrt{x ^2+9}$$
$$y=-\sqrt{x ^2+9}$$

marigirl · Answer

I gotta go now, sorry i can come back later and solve it by hand .i have a feeling the ans could be irrational

marigirl · Answer

and plz give me a medal  for the first one :)

anonymous · Answer

Please come back later! I will give you medals if you can help me solve just this last one :)

marigirl · Answer

I got it :D :D

marigirl · Answer

I first took the circle equation x^2+y^2-16y+39=0 and make it as 
$$y=8-\sqrt{25-x ^{2}}$$ then Intersect with $$y=\sqrt{x^2+9}$$

marigirl · Answer

so it looks like this

marigirl · Answer

so the 3 points of intersection are (-4,5) (0,3) and (4,5)