SOLUTION: line Z goes through the point (2, -4) and is perpendicular to the line through (-6,5) and (-3, -2). What is the slope of Z? I don't even know where to start.

Question 98508: line Z goes through the point (2, -4) and is perpendicular to the line through (-6,5) and (-3, -2). What is the slope of Z? I don't even know where to start.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Lets find the equation of the line through (-6,5) and (-3, -2)

First lets find the slope through the points (,) and (,)

Start with the slope formula (note:

is the first point (,) and

is the second point (,))

Plug in ,,, (these are the coordinates of given points)

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

So the slope is

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Now let's use the point-slope formula to find the equation of the line:

------Point-Slope Formula------
where is the slope, and

is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)

Rewrite as

Distribute

Multiply and to get . Now reduce to get

Add to both sides to isolate y

Combine like terms and to get
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Answer:

So the equation of the line which goes through the points (,) and (,) is:

Now let's find the equation of the line perpendicular to and goes through (2,-4)

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line

Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of , you can find the perpendicular slope by this formula:

where is the perpendicular slope

So plug in the given slope to find the perpendicular slope

When you divide fractions, you multiply the first fraction (which is really ) by the reciprocal of the second

Multiply the fractions.

So the perpendicular slope is

So now we know the slope of the unknown line is (its the negative reciprocal of from the line ). Also since the unknown line goes through (2,-4), we can find the equation by plugging in this info into the point-slope formula

Point-Slope Formula:

where m is the slope and (,) is the given point

Plug in , , and

Distribute

Multiply

Subtract from both sides to isolate y

Make into equivalent fractions with equal denominators

Combine the fractions

Reduce any fractions

So the equation of the line that is perpendicular to and goes through (,) is

So here are the graphs of the equations and

graph of the given equation (red) and graph of the line (green) that is perpendicular to the given graph and goes through (,)

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