Answer:
[tex]y = 13\, x + 5[/tex].
Step-by-step explanation:
Start by finding the slope of this line.
In general, if a non-vertical line contains two points, [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex], where [tex]x_{0} \ne x_{1}[/tex], the slope [tex]m[/tex] of this line would be:
[tex]\displaystyle \frac{y_{1} - y_{0}}{x_{1} - x_{0}}[/tex].
In this question, the two points on this line are [tex](6,\, 83)[/tex] and [tex](9,\, 122)[/tex]. Therefore, the slope of this line would be:
[tex]\begin{aligned}& \frac{122 - 83}{9 - 6} \\ =\; & \frac{39}{3} \\ =\; & 13\end{aligned}[/tex].
Next, find the equation of this line in the point-slope form.
In general, if a non-vertical line of slope [tex]m[/tex] contains a point [tex](x_{0},\, y_{0})[/tex], the point-slope form of this line would be:
[tex]\displaystyle y - y_{0} = m\, (x - x_{0})[/tex].
In this question, it was already found that the slope [tex]m[/tex] takes the value [tex]13[/tex]. Two points on this line are given. Using either of them as [tex](x_{0},\, y_{0})[/tex] would work. For example, with [tex](6,\, 83)[/tex] as the point, the point-slope equation of this line would be:
[tex]y - 83 = 13\, (x - 6)[/tex].
Finally, rewrite the point-slope form equation of this line into the slope-intercept form.
The slope-intercept equation of a line is of the form [tex]y = m\, x + b[/tex] for slope [tex]m[/tex] and [tex]y[/tex]-intercept [tex]b[/tex]. Rearrange the point-slope equation [tex]y - 83 = 13\, (x - 6)[/tex] into this new form:
[tex]y = 13\, (x - 6) + 83[/tex].
[tex]y = 13\, x - 78 + 83[/tex].
[tex]y = 13\, x + 5[/tex].
