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Write an equation in point slope form of the line that passes through the two

points given. (9.-2) and (-3, 2)

User Gadoma
by
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1 Answer

7 votes

Answer:


\displaystyle (y + 2) = -(1)/(3)\, (x - 9).

Equivalently:


\displaystyle (y - 2) = -(1)/(3)\, (x + 3).

Explanation:

Let
(x_(1),\, y_(1)) and
(x_(2)\, y_(2)) denote the coordinates of two points in the plane. If
x_(1) \\e x_(2) (i.e., the two points aren't on the same vertical line,) the slope of the line that goes through the two points would be:


\begin{aligned} m = (y_(2) - y_(1))/(x_(2) - x_(1))\end{aligned}.

In this question,
x_(1) = 9 and
y_(1) = (-2) (for
(9,\, -2)) while
x_(2) = (-3) and
y_(2) = 2 (for
(-3,\, 2).) Given that
x_(1) \\e x_(2), the slope of the line that goes through these two points would be:


\begin{aligned} m &= (y_(2) - y_(1))/(x_(2) - x_(1)) \\ &= (2 - (-2))/((-3) - 9) \\ &= -(1)/(3)\end{aligned}

If a line in a plane is of slope
m and goes through the point
(x_(0),\, y_(0)), the point-slope equation of this line would be:


(y - y_(0)) = m\, (x - x_(0)).

The slope of this line is
m = (-1/3). If the point
(9,\, -2) is chosen as
(x_(0),\, y_(0)) (
x_(0) = 9 and
y_(0) = (-2),) then the point-slope equation of this line would be:


(y - (-2)) = (-1/3)\, (x - 9).

Simplify to obtain:


\displaystyle (y + 2) = -(1)/(3)\, (x - 9).

Similarly, if the point
(-3,\, 2) is chosen as
(x_(0),\, y_(0)), the point-slope equation of this would be:


\displaystyle (y - 2) = -(1)/(3)\, (x + 3).

User Nebuto
by
8.3k points

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