Question 80
<pre><font color = "blue"><b>Find the equation of the line passing through (1/2, 3/4) and is
perpendicular to {{{3x+5y=7}}}
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We need to know:

1.  That if the equation of a line is solved for y, and written in the
slope-intercept form, y = mx + b, then the slope will be the coefficient,
m, of x [also the y-intercept will be the point (0,b), but we won't need 
that here.]

2. That if we know the slope of one line, then the slope of a line
perpendicular to it will be found by taking the reciprocal of the slope and
changing its sign.

3.  That if we know the slope, m, of a line and a point (x<sub>1</sub>, y<sub>1</sub>) it passes
through, that its equation is given by the point-slope form y - y<sub>1</sub> = m(x - x<sub>1</sub>)
 
First we find the slope of the line whose equation is 3x+5y=7, by solving it
for y:

                           {{{3x + 5y = 7}}}
                                {{{5y = -3x + 7}}}

Divide the coefficient of every term by the coefficient of y, namely 5:

                                 {{{y = (-3/5)x + 7/5}}}

This is in the slope-intercept form y = mx + b.  So its slope is the
coefficient of x, namely -3/5.

Now to get the slope of a line perpendicular to the given line, we form the
reciprocal of -3/5, namely -5/3, and change its sign to +, and we get 5/3 for
the slope of a line perpendicular to the given line.

So now we have m = 5/3.

We are given a point it passes through, namely (1/2, 3/4), so x<sub>1</sub> = 1/2 and
y<sub>1</sub> = 3/4, so we substitute those into the slope-intercept equation

                       y - y<sub>1</sub> = m(x - x<sub>1</sub>)

                                 {{{y - 3/4 = (5/3)(x - 1/2)}}}

                                 {{{y - 3/4 = (5/3)x - (5/3)(1/2)}}}

                                 {{{y - 3/4 = (5/3)x - 5/6}}}

                                       {{{y = (5/3)x - 5/6 + 3/4}}}

Get a common denominator of 12 to combine the last two fractions:

                                       {{{y = (5/3)x - 10/12 + 9/12}}}

                                       {{{y = (5/3)x - 1/12}}}

That's the required equation.  However, you may be asked for the answer in
general form Ax + By = C.  [Note the capital "B" is not the same as the
small "b"].  

So we clear of fractions by multiplying through by 12

                                     {{{12y = 12(5/3)x - 12(1/12)}}}

                                     {{{12y = 20x - 1}}} 

                              {{{-20x + 12y = -1}}}

It is usually preferred that the general form doesn't begin with a negative
sign, so we multiply through by -1 to change all the signs:

                               {{{20x - 12y = 1}}} 

That's the answer!

Edwin  <font face = "wingdings" size = 7 color = "red">J</font>