8A F11 Q15

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, y_1)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_2, y_2)} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{y_2 - y_1}{x_2 - x_1}} .
2) The point-slope form of a line is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - y_1 = m (x - x_1)} where the slope of the line is m, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{-1}{m}}

Solution:

Step 1:

Since the slope of a line passing through two points is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{y_2 - y_1}{x_2 - x_1} } , the slope of the line is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - 6 = 4(x - 5)} . Another answer is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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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