Answer:
[tex](y - 9) = (-8/3)\, (x - 7)[/tex].
Step-by-step explanation:
If a line in a cartesian plane has slope [tex]m[/tex], and the point [tex](x_{0},\, y_{0})[/tex] is on this line, then the point-slope equation of this line will be [tex](y - y_{0}) = m\, (x - x_{0})[/tex].
The slope of a line measures the rate of change in [tex]y[/tex]-coordinates relative to the change in the [tex]x[/tex]-coordinates. If a line goes through two points [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex], the slope of that line will be:
[tex]\begin{aligned}m &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}}\end{aligned}[/tex].
In this question, the two points on this line are [tex](7,\, 9)[/tex] and [tex](10,\, 1)[/tex], such that [tex]x_{0} = 7[/tex], [tex]y_{0} = 9[/tex], [tex]x_{1} = 10[/tex], and [tex]y_{1} = 1[/tex]. Substitute these values into the equation to find the slope of this line:
[tex]\begin{aligned}m &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}} \\ &= \frac{1 - 9}{10 - 7} \\ &= \left(-\frac{8}{3}\right)\end{aligned}[/tex].
With the point [tex](7,\, 9)[/tex] as the specific point [tex](x_{0},\, y_{0})[/tex] (such that [tex]x_{0} = 7[/tex] and [tex]y_{0} = 1[/tex]) as well as a slope of [tex]m = (-8 / 3)[/tex], the point-slope equation of this line will be:
[tex]y - y_{0} = m\, (x - x_{0})[/tex].
[tex]\displaystyle y - 9 = \left(-\frac{8}{3}\right)\, (x - 7)[/tex].
