004 Sample Final A, Problem 14

a) Find an equation of the line passing through (-4, 2) and (3, 6).
b) Find the slope of any line perpendicular to your answer from a)

Foundations
1) How do you find the slope of a line through points $(x_{1},y_{1})$ and $(x_{2},y_{2})$?
2) What is the equation of a line?
3) How do you find the slope of a line perpendicular to a line $y$?
Answer:
1) The slope is given by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}} .
2) The equation of a line is $y-y_{1}=m(x-x_{1})$ where $(x_{1},y_{1})$ is a point on the line.
3) The slope is given by $-{\frac {1}{m}}$ where $m$ is the slope of the line $y$.

Solution:

Step 1:
Using the above equation, the slope is equal to $m={\frac {6-2}{3-(-4)}}={\frac {4}{7}}$.

Step 2:
The equation of the line is $y-6={\frac {4}{7}}(x-3)$. Solving for 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} ,
we get 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=\frac{4}{7}x+\frac{30}{7}} .

Step 3:
The slope of any line perpendicular to the line in Step 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{1}{(\frac{4}{7})}=-\frac{7}{4}} .

Final Answer:
The slope 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{4}{7}} , the equation 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 y=\frac{4}{7}x+\frac{30}{7}} , and
the slope of any line perpendicular to this 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{7}{4}} .

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