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Math Help - Equation of the Line

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    Equation of the Line

    Show that an equation for a line with nonzero x-and y-intercepts can be written as (x/a) + (y/b) = 1 where a is the x-intercept and
    b is the y-intercept. This is called the INTERCEPT FORM of the equation of a line.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by symmetry View Post
    Show that an equation for a line with nonzero x-and y-intercepts can be written as (x/a) + (y/b) = 1 where a is the x-intercept and
    b is the y-intercept. This is called the INTERCEPT FORM of the equation of a line.
    The x-intercept will be of the form (a, 0). Does this fit the equation?
    \frac{a}{a} + \frac{0}{b} = 1 Check!

    The y-intercept will be of the form (0, b). Does this fit the equation?
    \frac{0}{a} + \frac{b}{b} = 1 Check!

    If you have a need to show that this equation is indeed a line, then you can solve it for y:
    \frac{x}{a} + \frac{y}{b} = 1

    \frac{y}{b} = -\frac{x}{a} + 1

    y = - \left ( \frac{b}{a} \right ) x + b

    which is the slope-intercept form for a line.

    -Dan
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    ok

    So, I simply needed to set y = 0 to find the x-intercepts and set x = 0 to find the y-intercepts.

    How about that?

    Thanks!
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by symmetry View Post
    So, I simply needed to set y = 0 to find the x-intercepts and set x = 0 to find the y-intercepts.

    How about that?

    Thanks!
    Yup! What a lot of students tend to forget is that the y - intercept, for example, is a point. So when we say the y - intercept is "b" we are using a short-hand and being a bit sloppy: we really should be saying the y - intercept is the point (0, b). The same goes for the x - intercept.

    -Dan
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    Hello, symmetry!

    You may be expect to derive that formula . . .


    Show that an equation for a line with nonzero x-and y-intercepts
    can be written as: . \frac{x}{a} + \frac{y}{b}\: = \:1
    where a is the x-intercept and b is the y-intercept.

    We are given two points on the line: . (a,0) and (0,b).

    The slope of the line is: . m \:=\:\frac{b-0}{0-a} \:=\:-\frac{b}{a}

    The line through (0,b) with slope -\frac{b}{a} is:
    . . . . y - b \:=\:-\frac{b}{a}(x - 0)\quad\Rightarrow\quad y \:=\:-\frac{b}{a}x + b\quad\Rightarrow\quad \frac{bx}{a} + y \:=\:b

    Divide through by b\!:\;\;\boxed{\frac{x}{a} + \frac{y}{b}\:=\:1}

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