IGreen's answer is elegant and totally correct, but I'm guessing that you'll get
confused when you try to dig through it. Allow me to present a simpler method,
if I may:
-- In your picture, we can only see 1, 3, and 5. We can't see enough of 2, 4,
or 6 to work on them.
But ALL six of the problems on this page are examples of the same thing:
Write the equation of a line when you know the slope and one point on the
line. That's it !
Rules to use:
-- The slope-intercept form of the equation of a line is
y = (slope) x + (intercept)
-- When two lines are parallel, they have the SAME slope.
-- When two lines are perpendicular, their slopes are negative reciprocals.
. . . . . 'negative' means the opposite sign
. . . . . 'reciprocal' means flip a fraction over, or take 1 / (a whole number).
negative reciprocal of 3/4 is (- 4/3)
negative reciprocal of -2 is (+ 1/2)
etc.
And now you're ready to solve all six problems.
#1). through (2,2) and parallel to [ y = x + 4 ] .
"parallel" means the new line has the same slope as [ y = x + 4 ] .
What is the slope of [ y = x + 4 ] ? It's the '1' that's not written next to 'x'.
So the new line is going to be [ y = 1x + b ].
All you need to find is 'b'.
Whatever equation you find for the new line, it'll be true for every point on
the line.
You already know one point on the new line ... (2,2) ... The equation must be
true for this point. Write this point into the new equation:
y = 1x + b
2 = 1(2) + b
2 = 2 + b
What is 'b' ? Subtract 2 from each side, and you have
0 = b .
So the equation of the new line is [ y = x + 0 ] or just [ y = x ].
That's how to do them when the new line is parallel to a given line.
Now do one where the new line is perpendicular to a given line.
#5). through (1, -5) and perpendicular to [ y = -1/8 x + 2 ].
What's the slope of the new line ? In order to be perpendicular to this one,
the slope of the new one must be the negative reciprocal of -1/8 .
The negative of -(anything) is +.
The reciprocal of 1/8 is 8 .
So the slope of the new line is +8 .
The equation of the new line is . . . [ y = 8x + b ].
How to find 'b' ?
Take the point you know on the new line (1, -5), and write it into this equation:
-5 = 8(1) + b
Squeeze 'b' out of this, and you have the equation of your new line.
All of the other problems on the page are just like these. All together,
on the whole page, there are 4 problems where the new line is parallel
to the given line, and two problems where the new line is perpendicular
to the given one. You should be able to whip out all six in about 15 minutes.
Good luck.
