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sophadof

  • one year ago

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

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  1. anonymous
    • one year ago
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    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.

  2. jtvatsim
    • one year ago
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    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).

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

  4. jtvatsim
    • one year ago
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    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?

  5. jtvatsim
    • one year ago
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    Of course, there is still work to do from where I left off, but hopefully that gets you started. :)

  6. jtvatsim
    • one year ago
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    You can check your answer against this graph I did on Desmos.com

  7. sophadof
    • one year ago
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    I am completely lost :( @jtvatsim

  8. sophadof
    • one year ago
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    it makes sense..it's just moving on from what you did is losing me.. @jtvatsim

  9. jtvatsim
    • one year ago
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    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?

  10. sophadof
    • one year ago
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    yes!!(: @jtvatsim i actualy did haha

  11. jtvatsim
    • one year ago
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    OK, great! Now, let's subtract 25 from both sides leaving us with: 2x^2 - 10x = 0

  12. jtvatsim
    • one year ago
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    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.

  13. jtvatsim
    • one year ago
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    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! :)

  14. jtvatsim
    • one year ago
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    Any questions? :)

  15. sophadof
    • one year ago
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    haha hang on..just gonna try it(: @jtvatsim

  16. jtvatsim
    • one year ago
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    Sounds good.

  17. sophadof
    • one year ago
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    Thank you for staying in this with me though!!!!(: @jtvatsim

  18. jtvatsim
    • one year ago
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    No worries, glad to help if I can. :)

  19. sophadof
    • one year ago
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    yeah i got x=5 @jtvatsim

  20. jtvatsim
    • one year ago
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    And you should also get x = 0 as another solution.

  21. sophadof
    • one year ago
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    oh! @jtvatsim gotcha!

  22. jtvatsim
    • one year ago
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    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. :)

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

  24. jtvatsim
    • one year ago
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    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. :)

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

  26. sophadof
    • one year ago
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    @jtvatsim

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  27. jtvatsim
    • one year ago
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    And we are to find the value of x correct?

  28. sophadof
    • one year ago
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    yes

  29. jtvatsim
    • one year ago
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    Alright, first question, do you know what the ~ symbol means?

  30. sophadof
    • one year ago
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    similar right?

  31. jtvatsim
    • one year ago
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    Correct!

  32. jtvatsim
    • one year ago
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    So, that basically means the triangles have the same shape (maybe not the same size).

  33. jtvatsim
    • one year ago
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    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.

  34. sophadof
    • one year ago
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    haha thank you

  35. jtvatsim
    • one year ago
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    |dw:1433640781894:dw| That's better.

  36. jtvatsim
    • one year ago
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    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?

  37. sophadof
    • one year ago
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    yes!

  38. jtvatsim
    • one year ago
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    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). :)

  39. sophadof
    • one year ago
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    ill choose 75

  40. jtvatsim
    • one year ago
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    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?

  41. sophadof
    • one year ago
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    4x-10

  42. jtvatsim
    • one year ago
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    Perfect!

  43. jtvatsim
    • one year ago
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    We now just repeat the process for the other sides we know. We should get something like this...

  44. jtvatsim
    • one year ago
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    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?

  45. sophadof
    • one year ago
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    nope im good!!

  46. jtvatsim
    • one year ago
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    Very good! :)

  47. jtvatsim
    • one year ago
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    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)

  48. jtvatsim
    • one year ago
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    Very important step right there, are you ok with that?

  49. sophadof
    • one year ago
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    yes! then do i solve?

  50. jtvatsim
    • one year ago
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    Yes! Now you have a "nice" looking equation (you do have some distributing to do, so be careful).

  51. sophadof
    • one year ago
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    so...75x+375=120x-300?

  52. jtvatsim
    • one year ago
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    Yes, that is correct!

  53. sophadof
    • one year ago
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    yay!!!!!!!!!(: thank you!

  54. jtvatsim
    • one year ago
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    no problem!

  55. sophadof
    • one year ago
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    then subtract 75 from 120?

  56. jtvatsim
    • one year ago
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    Sure, that leaves you with 45x I believe.

  57. sophadof
    • one year ago
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    x=15?(:

  58. jtvatsim
    • one year ago
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    You got it!

  59. sophadof
    • one year ago
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    yay!!!!! Thank you soooo much!!(: do i ned to plug in x for those equations or no?

  60. jtvatsim
    • one year ago
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    Only if the question wanted you to find the actual sides of the unknown triangle. Sometimes they do want you to find that.

  61. sophadof
    • one year ago
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    oh alrighty!!! thank you so much!!

  62. jtvatsim
    • one year ago
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    That's easy to find though based on the information we now have.

  63. jtvatsim
    • one year ago
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    Your welcome!

  64. sophadof
    • one year ago
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    Have a good night? day? haha

  65. jtvatsim
    • one year ago
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    More like evening. :) You too!

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