sophadof
  • sophadof
Find the point(s) of intersection of the line X + Y = 5 and the circle x2 + y2 = 25 by solving the system of equations.
Geometry
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jamiebookeater
  • jamiebookeater
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anonymous
  • 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
  • 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
  • sophadof
wait..so @jtvatsim after i plug it in the step after would be x+(5)^2=25?

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jtvatsim
  • 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
  • jtvatsim
Of course, there is still work to do from where I left off, but hopefully that gets you started. :)
jtvatsim
  • jtvatsim
You can check your answer against this graph I did on Desmos.com
sophadof
  • sophadof
I am completely lost :( @jtvatsim
sophadof
  • sophadof
it makes sense..it's just moving on from what you did is losing me.. @jtvatsim
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
  • sophadof
yes!!(: @jtvatsim i actualy did haha
jtvatsim
  • jtvatsim
OK, great! Now, let's subtract 25 from both sides leaving us with: 2x^2 - 10x = 0
jtvatsim
  • 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
  • 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
  • jtvatsim
Any questions? :)
sophadof
  • sophadof
haha hang on..just gonna try it(: @jtvatsim
jtvatsim
  • jtvatsim
Sounds good.
sophadof
  • sophadof
Thank you for staying in this with me though!!!!(: @jtvatsim
jtvatsim
  • jtvatsim
No worries, glad to help if I can. :)
sophadof
  • sophadof
yeah i got x=5 @jtvatsim
jtvatsim
  • jtvatsim
And you should also get x = 0 as another solution.
sophadof
  • sophadof
oh! @jtvatsim gotcha!
jtvatsim
  • 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
  • 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
  • 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
yes thank you!!!!! @jtvatsim I hate to ask you this but...do you have email, I have another math question and its with triangles..
sophadof
  • sophadof
@jtvatsim
1 Attachment
jtvatsim
  • jtvatsim
And we are to find the value of x correct?
sophadof
  • sophadof
yes
jtvatsim
  • jtvatsim
Alright, first question, do you know what the ~ symbol means?
sophadof
  • sophadof
similar right?
jtvatsim
  • jtvatsim
Correct!
jtvatsim
  • jtvatsim
So, that basically means the triangles have the same shape (maybe not the same size).
jtvatsim
  • 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
  • sophadof
haha thank you
jtvatsim
  • jtvatsim
|dw:1433640781894:dw| That's better.
jtvatsim
  • 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
  • sophadof
yes!
jtvatsim
  • 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
  • sophadof
ill choose 75
jtvatsim
  • 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
  • sophadof
4x-10
jtvatsim
  • jtvatsim
Perfect!
jtvatsim
  • jtvatsim
We now just repeat the process for the other sides we know. We should get something like this...
jtvatsim
  • 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
  • sophadof
nope im good!!
jtvatsim
  • jtvatsim
Very good! :)
jtvatsim
  • 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
  • jtvatsim
Very important step right there, are you ok with that?
sophadof
  • sophadof
yes! then do i solve?
jtvatsim
  • jtvatsim
Yes! Now you have a "nice" looking equation (you do have some distributing to do, so be careful).
sophadof
  • sophadof
so...75x+375=120x-300?
jtvatsim
  • jtvatsim
Yes, that is correct!
sophadof
  • sophadof
yay!!!!!!!!!(: thank you!
jtvatsim
  • jtvatsim
no problem!
sophadof
  • sophadof
then subtract 75 from 120?
jtvatsim
  • jtvatsim
Sure, that leaves you with 45x I believe.
sophadof
  • sophadof
x=15?(:
jtvatsim
  • jtvatsim
You got it!
sophadof
  • sophadof
yay!!!!! Thank you soooo much!!(: do i ned to plug in x for those equations or no?
jtvatsim
  • 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
  • sophadof
oh alrighty!!! thank you so much!!
jtvatsim
  • jtvatsim
That's easy to find though based on the information we now have.
jtvatsim
  • jtvatsim
Your welcome!
sophadof
  • sophadof
Have a good night? day? haha
jtvatsim
  • jtvatsim
More like evening. :) You too!

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