008A Sample Final A, Question 15

Question: a) Find the equation of the line passing through (3, -2) and (5, 6).
                  b) Find the slope of any line perpendicular to your answer from a)

Foundations
1) We have two points on a line. How do we find the slope?
2) How do you write the equation of a line, given a point on the line and the slope?
3) For part b) how are the slope of a line and the slope of all perpendicular lines related?
Answer:
1) The formula for the slope of a line through two points $(x_{1},y_{1})$  and  $(x_{2},y_{2})$   is  ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$.
2) The point-slope form of a line is $y-y_{1}=m(x-x_{1})$ where the slope of the line is m, and $(x_{1},y_{1})$ is a point on the line.
3) If m is the slope of a line. The slope of all perpendicular lines is ${\frac {-1}{m}}$

Solution:

Step 1:
Since the slope of a line passing through two points is ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$, the slope of the line is ${\frac {6-(-2)}{5-3}}={\frac {8}{2}}=4$

Final Answer part a):
Now that we have the slope of the line and a point on the line the equation for the line is $y-6=4(x-5)$. Another answer is $y+2=4(x-3)$. These answers are the same. They just look different.

Final Answer part b):
Since the slope of the line in part a) is 4, the slope of any line perpendicular to the answer in a) is ${\frac {-1}{4}}$

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