# 004 Sample Final A, Problem 14

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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 ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$?
2) What is the equation of a line?
3) How do you find the slope of a line perpendicular to a line ${\displaystyle y}$?
1) The slope is given by ${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$.
2) The equation of a line is ${\displaystyle y-y_{1}=m(x-x_{1})}$ where ${\displaystyle (x_{1},y_{1})}$ is a point on the line.
3) The slope is given by ${\displaystyle -{\frac {1}{m}}}$ where ${\displaystyle m}$ is the slope of the line ${\displaystyle y}$.

Solution:

Step 1:
Using the above equation, the slope is equal to ${\displaystyle m={\frac {6-2}{3-(-4)}}={\frac {4}{7}}}$.
Step 2:
The equation of the line is ${\displaystyle y-6={\frac {4}{7}}(x-3)}$. Solving for ${\displaystyle y}$,
we get ${\displaystyle y={\frac {4}{7}}x+{\frac {30}{7}}}$.
Step 3:
The slope of any line perpendicular to the line in Step 2 is ${\displaystyle -{\frac {1}{({\frac {4}{7}})}}=-{\frac {7}{4}}}$.
The slope is ${\displaystyle {\frac {4}{7}}}$, the equation of the line is ${\displaystyle y={\frac {4}{7}}x+{\frac {30}{7}}}$, and
the slope of any line perpendicular to this line is ${\displaystyle -{\frac {7}{4}}}$.