Answer:
[tex]slope = \frac{y_2 - y_1}{x_2-x_1}\\\\4 = \frac{p-2}{1-4} \\\\4 \times -3 = p -2\\\\-12 + 2 = p \\\\p = -10 \\\\Equation \ of \ line : \\\\(y -y_1) = m(x - x_1)\\\\y - 2= 4(x-4)\\\\y - 2 = 4x -16\\\\y = 4x -14[/tex]
Hi there! The slope formula is:
[tex] \large \boxed{m = \frac{y_2 - y_1}{x_2 - x_1} }[/tex]
We know that the slope is 4 but we are missing the value of p. (Define m = slope)
What we have now are:
We are going to substitute these values and solve the equation for p-term.
[tex] \large{4 = \frac{2 - p}{4 - 1} } \\ \large{4 = \frac{2 - p}{3} } \\ \large{4(3) = \frac{2 - p}{3} (3)} \\ \large{12 = 2 - p} \\ \large{p = 2 - 12 \longrightarrow \boxed{p = - 10}}[/tex]
Hence, the value of p is -10.
Next, we have to write the equation of a line in point-slope form. The point-slope form is:
[tex] \large \boxed{y - y_1 = m(x - x_1)}[/tex]
Define that (x1, y1) = ordered pairs
Since we have two given points, we can either use the first point or second point. Both work.
First Point
For our first ordered pairs (4,2), substitute x1 = 4 and y1 = 2 in the equation.
[tex] \large{y - 2 = 4(x - 4)}[/tex]
Hence, the equation of a line in point-slope form as in (4,2) is y-2=4(x-4)
Second Point
For our second ordered pairs (1,p), we know that p = -10 from the equation that we solved. Therefore (1,p) = (1,-10). Substitute x1 = 1 and y1 = -10 in the equation.
[tex] \large{y - ( - 10) = 4(x - 1)} \\ \large{y + 10 = 4(x - 1)}[/tex]
Hence, the equation of a line in point-slope form as in (1,-10) is y+10 = 4(x-1)
Answer
Both equations work for point-slope form.
Questions can be asked through comment.
Hope this helps, and Happy Learning! :)
