Find the point(s) of intersection of the line X + Y = 5 and the circle x2 + y2 = 25 by solving the system of equations.

- sophadof

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- anonymous

My geometry is very rusty, but I'll give it a shot...
If I remember correctly, you will need to put both equations into slope-intercept forum. Doing this, you will get y= -x+5 and y= x+5 ( someone may want to double check this). After this, you use trace the two equations and will get an intersection of y=5.
Again, my geometry is rusty and someone may want to double check.

- jtvatsim

Iehmad is close. The idea is to solve the first equation for y.
1) This gives y = -x + 5.
2) You will then plug this into the second equation x^2 + y^2 = 25,
to get x^2 + (-x + 5)^2 = 25.
3) Solve this for x.
4) You can then use the x's you find to solve for y.
5) Your final answer will be coordinates (x,y).

- sophadof

wait..so @jtvatsim after i plug it in the step after would be x+(5)^2=25?

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- jtvatsim

The second equation is x^2 + y^2 = 25. We will replace y with y = -x + 5. That makes the equation: x^2 + (-x + 5)^2 = 25. Be careful to factor the entire parentheses.
You should get x^2 + (-x + 5)*(-x + 5) = 25 which simplifies to
x^2 + (x^2 - 5x - 5x + 25) = 25,
x^2 + x^2 - 10x + 25 = 25.
Does that make sense?

- jtvatsim

Of course, there is still work to do from where I left off, but hopefully that gets you started. :)

- jtvatsim

You can check your answer against this graph I did on Desmos.com

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- sophadof

I am completely lost :( @jtvatsim

- sophadof

it makes sense..it's just moving on from what you did is losing me.. @jtvatsim

- jtvatsim

No problem. Here's how I would do it. After simplifying the work I did above you should get:
2x^2 - 10x + 25 = 25. Did you get that?

- sophadof

yes!!(: @jtvatsim i actualy did haha

- jtvatsim

OK, great! Now, let's subtract 25 from both sides leaving us with:
2x^2 - 10x = 0

- jtvatsim

I can factor out a 2 (not essential, but why not?) to get
2(x^2 - 5x) = 0
then,
x^2 - 5x = 0 (after dividing by 2).
Factoring, gives me
x(x - 5) = 0.
so x = 0, or x = 5.

- jtvatsim

Now WAY BACK in the beginning, we said that y = -x + 5. So we can plug in our new found values of x to get the y values.
x = 0, means y = -0 + 5 = 5. This is the point (0,5).
x = 5, means y = -5 + 5 = 0. This is the point (5,0).
We are done. Yay! :)

- jtvatsim

Any questions? :)

- sophadof

haha hang on..just gonna try it(: @jtvatsim

- jtvatsim

Sounds good.

- sophadof

Thank you for staying in this with me though!!!!(: @jtvatsim

- jtvatsim

No worries, glad to help if I can. :)

- sophadof

yeah i got x=5 @jtvatsim

- jtvatsim

And you should also get x = 0 as another solution.

- sophadof

oh! @jtvatsim gotcha!

- jtvatsim

Great! If you had done something like x^2 = 5x and divided by x, you would get x = 5, but dividing by x accidentally causes you to miss the x = 0 solution. Just wanted to be sure you saw that. :)

- sophadof

yes thank you!!!!! @jtvatsim I hate to ask you this but...do you have email, I have another math question and its with triangles..

- jtvatsim

No worries, feel free to post it on OpenStudy just in case I have to run here soon. At least someone else would be able to jump in. :)

- sophadof

- sophadof

@jtvatsim

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- jtvatsim

And we are to find the value of x correct?

- sophadof

yes

- jtvatsim

Alright, first question, do you know what the ~ symbol means?

- sophadof

similar right?

- jtvatsim

Correct!

- jtvatsim

So, that basically means the triangles have the same shape (maybe not the same size).

- jtvatsim

The problem is that the picture didn't really do a good job at showing this (especially since one is upside down!). I'll go ahead and fix that.

- sophadof

haha thank you

- jtvatsim

|dw:1433640781894:dw|
That's better.

- jtvatsim

Now, whenever we have similar triangles, we like to notice the sides that match and set up a "proportion". Something like these fractions:
(big triangle side 1)/(big triangle side 2) = (small triangle side 1)/(small triangle side 2)
Make sense so far?

- sophadof

yes!

- jtvatsim

Cool! So, now let's pick some sides! What would you like to pick for "big triangle side 1"? You can pick whatever you like (based on the drawing of course). :)

- sophadof

ill choose 75

- jtvatsim

OK, so right now we have:
75/(big triangle side 2) = (small triangle side 1)/(small triangle side 2).
We need to keep going though. I would suggest that we pick "small triangle side 1" now. We need to pick the same part of the smaller triangle. (In this case the bottom). What would that be for the small triangle?

- sophadof

4x-10

- jtvatsim

Perfect!

- jtvatsim

We now just repeat the process for the other sides we know. We should get something like this...

- jtvatsim

75/30 = (4x - 10)/(x + 5)
Looks a little scary, but we have now got the question ready to solve. Any questions so far before we go on?

- sophadof

nope im good!!

- jtvatsim

Very good! :)

- jtvatsim

Right now, cross multiplication is your friend (or if you prefer multiply each fraction by the bottoms) and this gives us:
75*(x + 5) = 30*(4x - 10)

- jtvatsim

Very important step right there, are you ok with that?

- sophadof

yes! then do i solve?

- jtvatsim

Yes! Now you have a "nice" looking equation (you do have some distributing to do, so be careful).

- sophadof

so...75x+375=120x-300?

- jtvatsim

Yes, that is correct!

- sophadof

yay!!!!!!!!!(: thank you!

- jtvatsim

no problem!

- sophadof

then subtract 75 from 120?

- jtvatsim

Sure, that leaves you with 45x I believe.

- sophadof

x=15?(:

- jtvatsim

You got it!

- sophadof

yay!!!!! Thank you soooo much!!(:
do i ned to plug in x for those equations or no?

- jtvatsim

Only if the question wanted you to find the actual sides of the unknown triangle. Sometimes they do want you to find that.

- sophadof

oh alrighty!!! thank you so much!!

- jtvatsim

That's easy to find though based on the information we now have.

- jtvatsim

Your welcome!

- sophadof

Have a good night? day? haha

- jtvatsim

More like evening. :) You too!

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