Question 358479
Which is the equation of a line parallel to a line containing the points (-3, 1) and (6, 4) and passing throught the point (5, 2)?
<pre> 
First we'll draw the points that the first line goes through:
 
{{{drawing(400,400,-9,9,-9,9,
locate(6,4,"(6,4)"), locate(-3,1,"(-3,1)"),
graph(400,400,-9,9,-9,9), 
 
line(-3+.1,1,-3-.1,1),line(-3,1+.1,-3,1-.1),line(-3+.1,1+.1,-3-.1,1-.1),line(-3+.1,1-.1,-3-.1,1+.1),
line(6+.1,4,6-.1,4),line(6,4+.1,6,4-.1),line(6+.1,4+.1,6-.1,4-.1),line(6+.1,4-.1,6-.1,4+.1) )}}}
 
And then we'll draw that line.
 
{{{drawing(400,400,-9,9,-9,9,
locate(6,4,"(6,4)"), locate(-3,1,"(-3,1)"),
 
graph(400,400,-9,9,-9,9), 
line(-12,-2, 12,6),
line(-3+.1,1,-3-.1,1),line(-3,1+.1,-3,1-.1),line(-3+.1,1+.1,-3-.1,1-.1),line(-3+.1,1-.1,-3-.1,1+.1),
line(6+.1,4,6-.1,4),line(6,4+.1,6,4-.1),line(6+.1,4+.1,6-.1,4-.1),line(6+.1,4-.1,6-.1,4+.1) )}}}
 
We'll find the slope of that line using
 
{{{m}}}{{{""=""}}}{{{(y[2]-y[1])/(x[2]-x[1])}}}{{{""=""}}}{{{((4)-(1))/((6)-(-3))}}}{{{""=""}}}{{{3/(6+3)}}}{{{""=""}}}{{{3/9}}}{{{""=""}}}{{{1/3}}}

The line parallel to that one must slope the same way, that is,
it must have the same slope.

 Now we'll use the point-slope formula with (x<sub>1</sub>,y<sub>1</sub>) = (5,2)

{{{y-y[1]}}}{{{""=""}}}{{{m(x-x[1])}}}

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

Multiply both sides by 3:

{{{3(y-2)}}}{{{""=""}}}{{{3*expr(1/3)(x-5)}}}

{{{3y-6}}}{{{""=""}}}{{{cross(3)*expr(1/cross(3))(x-5)}}}

{{{3y-6}}}{{{""=""}}}{{{x-5}}}

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

{{{x-3y}}}{{{""=""}}}{{{5}}}

To check, we'll plot the point (5,2) 

{{{drawing(400,400,-9,9,-9,9,
locate(6,4,"(6,4)"), locate(-3,1,"(-3,1)"),
 
graph(400,400,-9,9,-9,9), 
line(-12,-2, 12,6),
line(-3+.1,1,-3-.1,1),line(-3,1+.1,-3,1-.1),line(-3+.1,1+.1,-3-.1,1-.1),line(-3+.1,1-.1,-3-.1,1+.1),
line(6+.1,4,6-.1,4),line(6,4+.1,6,4-.1),line(6+.1,4+.1,6-.1,4-.1),line(6+.1,4-.1,6-.1,4+.1),

line(5+.1,2,5-.1,2),line(5,2+.1,5,2-.1),line(5+.1,2+.1,5-.1,2-.1),line(5+.1,2-.1,5-.1,2+.1), locate(5,2,"(5,2)") )}}}

Then we'll get a couple more points
 
 x | y
 2 | 1
-1 | 0
 

{{{drawing(400,400,-9,9,-9,9,
locate(6,4,"(6,4)"), locate(-3,1,"(-3,1)"),
 
graph(400,400,-9,9,-9,9), 
line(-12,-2, 12,6),
line(-3+.1,1,-3-.1,1),line(-3,1+.1,-3,1-.1),line(-3+.1,1+.1,-3-.1,1-.1),line(-3+.1,1-.1,-3-.1,1+.1),
line(6+.1,4,6-.1,4),line(6,4+.1,6,4-.1),line(6+.1,4+.1,6-.1,4-.1),line(6+.1,4-.1,6-.1,4+.1),

line(5+.1,2,5-.1,2),line(5,2+.1,5,2-.1),line(5+.1,2+.1,5-.1,2-.1),line(5+.1,2-.1,5-.1,2+.1), locate(5,2,"(5,2)"),

line(2+.1,1,2-.1,1),line(2,1+.1,2,1-.1),line(2+.1,1+.1,2-.1,1-.1),line(2+.1,1-.1,2-.1,1+.1), locate(2,1,"(2,1)"),

line(-1+.1,0,-1-.1,0),line(-1,0+.1,-1,0-.1),line(-1+.1,0+.1,-1-.1,0-.1),line(-1+.1,0-.1,-1-.1,0+.1), locate(-1,0,"(-1,0)")

)}}}

Draw a line through those:
{{{drawing(400,400,-9,9,-9,9,
locate(6,4,"(6,4)"), locate(-3,1,"(-3,1)"),

graph(400,400,-9,9,-9,9), 
line(-12,-2, 12,6),
line(-3+.1,1,-3-.1,1),line(-3,1+.1,-3,1-.1),line(-3+.1,1+.1,-3-.1,1-.1),line(-3+.1,1-.1,-3-.1,1+.1),
line(6+.1,4,6-.1,4),line(6,4+.1,6,4-.1),line(6+.1,4+.1,6-.1,4-.1),line(6+.1,4-.1,6-.1,4+.1),

line(5+.1,2,5-.1,2),line(5,2+.1,5,2-.1),line(5+.1,2+.1,5-.1,2-.1),line(5+.1,2-.1,5-.1,2+.1), locate(5,2,"(5,2)"),

line(2+.1,1,2-.1,1),line(2,1+.1,2,1-.1),line(2+.1,1+.1,2-.1,1-.1),line(2+.1,1-.1,2-.1,1+.1), locate(2,1,"(2,1)"),

line(-1+.1,0,-1-.1,0),line(-1,0+.1,-1,0-.1),line(-1+.1,0+.1,-1-.1,0-.1),line(-1+.1,0-.1,-1-.1,0+.1), locate(-1,0,"(-1,0)"), green(line(-10,-3,11,4)) 

)}}}


So the green line is parallel to the black line and it goes through (5,2).

Edwin</pre>