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Definition: Let $P(x_0, y_0)$ be a point and let $m$ be a real number. The line through $P$ with slope $m$ is the set of all points $Q(x, y)$ with,

$y -y_0 = m(x - x_0)$

Does the set of all points $Q(x, y)$ also includes the point $P(x_0, y_0)$ since its on the line?

Martin Sleziak
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Samama Fahim
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2 Answers2

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Just set $x=x_0$, $y=x_0$. Your definition of line yields that $(x_0,y_0)$ lies on the line if and only if $$ y_0 - y_0 = m(x_0 - x_0) \text{.} $$ Which, of course, is true no matter the value of $x_0$, $y_0$ and $m$.

fgp
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Yes, indeed, Samama. Since we are creating the equation of the line using point P, we are assured that $P = (x_0, y_0)$ is on the line. Just substitute the coordinates of $P$ into the equation, and you will satisfy the equality given by the equation of the line. So $P$ is one of the points included on the line $L$ which is the set of all points determined by a given $m$, $P = (x_0, y_0)$: $$L = Q(x, y) = \{(x, y)\mid y -y_0 = m(x - x_0),\}$$

since $y_0 - y_0 = m(x_0 - x_0) \iff 0 = 0 \;\;\large \checkmark$

But $P$ is only one point in the infinite set defined by $Q(x, y)$. So don't mistaken $Q(x, y)$ as one point $Q$, it defines all points on the line determined by the given slope $m$, and a given point $P = (x_0, y_0)$ lying on the line.

amWhy
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    Please look at the following:

    If $1.5$ is the slope of a line and $y = 3$ $y_0 = 3$ also $x = 1$ $x_0 = 1$ then,

    $3 -3 = 1.5(1 -1)$ But, $\frac{3-3} {1-1} \neq 1.5$ so this doesn't satisfy the equation.

     – Samama Fahim Apr 29 '13 at 20:57
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    I'll be very thankful to you if you tell your opinion on my question in the link:

    http://math.stackexchange.com/questions/376227/point-slope-equation/376241?noredirect=1#comment805981_376241

     – Samama Fahim Apr 29 '13 at 21:02
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    Slope is 1.5: that's given. $3 - 3 = 0 = 1.5(1 - 1) = 1.5\times 0$. since $0 = 0$, $P = (x_0, y_O)$ is clearly on the line. However, you're correct, we can't determine a line's slope from just one point. If we aren't given slope, then we need two points to determine a line: With two distinct points, one can first determine slope, and then use one of the points to use the point slope form you've posted to find the equation of the line. – amWhy Apr 29 '13 at 21:04
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    @amWhy: okay, tried to upvote again! :-) – Amzoti May 01 '13 at 00:59