Find the equation of the line passing through (1/2, 3/4) and is
perpendicular to
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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 (x1, y1) it passes
through, that its equation is given by the point-slope form y - y1 = m(x - x1)
First we find the slope of the line whose equation is 3x+5y=7, by solving it
for y:
Divide the coefficient of every term by the coefficient of y, namely 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 x1 = 1/2 and
y1 = 3/4, so we substitute those into the slope-intercept equation
y - y1 = m(x - x1)
Get a common denominator of 12 to combine the last two fractions:
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
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:
That's the answer!
Edwin J