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Math Help - Find a Line (IF) passes through Specified Y Cordinates

  1. #1
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    Find a Line (IF) passes through Specified Y Cordinates

    I have two parallel lines in which the slopes of the line are just equal. But the only thing is Line 2 is bigger than Line 1

    Line 1 (30,78) (19,41)
    Line 2 (51,99) ( 25,20)

    I want to know if LINE 1 passes through Y1 and Y2 of Line1. In this case Y1 of Line 1 =78 and Y2 of Line 1 =41 Please give me the equation or the algorithm.

    I know that it passes but how to find it by an equation. Please advice. I'm not a math expert. Please explain me in a way that I understand. Give it to me in step by steps.

    Thanks
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  2. #2
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    To find the equation of a line, you can use the point-slope formula, which is: y-y_1=a(x-x_1) where a is the slope.

    So first we need to find the slope, which is a=\frac{y_2-y_1}{x_2-x_1}

    So substitute: a=\frac{41-78}{19-30}\quad\rightarrow\quad a=\frac{-37}{-11}\quad\rightarrow\quad a=\frac{37}{11}

    So now substitute into the equation of the line: y-y_1=a(x-x_1)\quad\rightarrow\quad y-78=\frac{37}{11}(x-30)

    And now we finish it off: \boxed{y=\frac{37}{11}(x-30)+78}

    note, that is the equation of the line, but most people want it in slope-intercept form, so I'll do that for you.

    y=\frac{37}{11}(x-30)+78

    y=\frac{37}{11}x-\frac{37}{11}30+78

    y=\frac{37}{11}x-\frac{1110}{11}+78

    y=\frac{37}{11}x-\frac{1110}{11}+\frac{858}{11}

    y=\frac{37}{11}x-\frac{1110+858}{11}

    \boxed{y=\frac{37}{11}x-\frac{1968}{11}}
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    Dear Quick

    Thank you very much for your QUICK reply. I'm very much happy with the way you have described the answer. But... I do have one more question left. Having the equation of the line how do I substitute the values and find of the Y cordinates of line 1 i,e 78 and 41 passes through line 2 ??????

    Thanks again
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    MHF Contributor Quick's Avatar
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    Quote Originally Posted by arunkish View Post
    Dear Quick

    Thank you very much for your QUICK reply. I'm very much happy with the way you have described the answer. But... I do have one more question left. Having the equation of the line how do I substitute the values and find of the Y cordinates of line 1 i,e 78 and 41 passes through line 2 ??????

    Thanks again
    I am confused by this question.

    Do you want to know if the line ever reaches those y-coordinates? if so then yes, because any line that isn't flat goes past every y-coordinate eventually.

    Do you want to know when the line reaches those y-coordinates? If so then tell me and I'll figure it out (or someone else will)

    Do you want to know if the line passes through the two points (30,78) and (19,41)? In which case it will not pass through both of them.
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    Dear Quick

    Do you want to know if the line passes through the two points (30,78) and (19,41)? In which case it will not pass through both of them.
    Yes. This is somewhat right. But I do want to know only if the Y co-ordinates are getting passed. FYI, it might not pass through X1 and X2 of the Line1 since Line2 is apart from Line1. Sorry if I'm wrong quick.

    Please see the attachment for more details.
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  6. #6
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by arunkish View Post
    Dear Quick



    Yes. This is somewhat right. But I do want to know only if the Y co-ordinates are getting passed.

    Please see the attachment for more details.
    I can't read the attachment, it pops up wth 0100090000037800000002001c000000000004000000030108 00050000000b0200000000050000000c026604be0604000000 2e0118001c000000fb021000070000000000bc020000000001 02022253797374656d0004be0600005a270000fc5b110004ee 8339e8711e000c020000040000002d01000004000000020101 001c000000fb02ceff00000000000090010000000004400012 54696d6573204e657720526f6d616e00000000000000000000 00000000000000040000002d01010005000000090200000002 0d000000320a2d0000000100040000000000bd066504203c16 00040000002d010000030000000000



    Anyway, As said before, if a line isn't flat (and your line isn't) then it will pass through all the y-coordinates. If your line is really a segment then it will still pass those two y-coordinates.
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  7. #7
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    Please check back now. Yes. you are right, but how do u find it by the formula. Please see attachment now. It's in DOC format.
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    Dear All, Sorry if I'm troubling you. I hope that I'm not confusing you much. Thanks in advance.
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  9. #9
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by arunkish View Post
    Please check back now. Yes. you are right, but how do u find it by the formula. Please see attachment now. It's in DOC format.
    I still can't read that thing. Anyway, so you want to find when it will hit those y-values?

    Well the first thing you need to do is find an equation for line 2 (I already showed you how to do that)

    The equation is: y=\frac{79}{26}x+\frac{1455}{26} (please check my work, I did it quite quickly)

    So now you have a "y" value and you want to find the "x" value, so solve for x:

    y=\frac{79}{26}x+\frac{1455}{26}\quad\rightarrow\q  uad y-\frac{1455}{26}=\frac{79}{26}x

    then divide:

    \left(y-\frac{1455}{26}\right)\frac{26}{79}=x

    then solve away:

    \frac{26}{79}y-\frac{1455\cdot\rlap{|}26}{\rlap{|}26\cdot 79}=x

    So then you get: x=\frac{26y}{79}-\frac{1455}{79}

    Now you can substitute:

    x=\frac{26\cdot 78}{79}-\frac{1455}{79}=\frac{2028}{79}-\frac{1455}{79}=\frac{2028-1455}{79}=\frac{573}{79}

    So line 2 passes 78 when x\approx 7.25

    Make sure to check my work. Also, now you know how to solve for one, you should be able to solve for the other one as well.
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  10. #10
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    Dear Quick and all,

    Yes. I realized that the line will definitely pass through the Y1 and Y2 of line one. But my actual question is. As you can see from the attachment. What will be the value of X cordinates ?? Please see attachment for more and modified information. Sorry for troubling you a lot.

    Thousands of thanks again.

    Regards
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