To determine the equation of the line that intersects the points [tex]\((8, -10)\)[/tex] and [tex]\((9, 4)\)[/tex] in point-slope form, follow these steps:
1. Find the slope ([tex]\(m\)[/tex]) of the line:
The slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the points [tex]\((8, -10)\)[/tex] and [tex]\((9, 4)\)[/tex]:
[tex]\[
m = \frac{4 - (-10)}{9 - 8} = \frac{4 + 10}{9 - 8} = \frac{14}{1} = 14
\][/tex]
2. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a line passing through a point [tex]\((x_1, y_1)\)[/tex] with slope [tex]\(m\)[/tex] is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, we are given the point [tex]\((8, -10)\)[/tex] and the slope [tex]\(14\)[/tex].
3. Substitute the given point and the slope into the point-slope form:
Using the point [tex]\((8, -10)\)[/tex]:
[tex]\[
y - (-10) = 14(x - 8)
\][/tex]
4. Simplify the equation:
Simplifying [tex]\(y - (-10)\)[/tex] gives:
[tex]\[
y + 10 = 14(x - 8)
\][/tex]
So, the equation of the line in point-slope form, using the point [tex]\((8, -10)\)[/tex], is:
[tex]\[
y + 10 = 14(x - 8)
\][/tex]
