Question 117317
 Using graph paper, graph the question and plot the given ordered pair. Then construct a perpendicular segment and find the distance from the point to the line. 
26. 3x + 4y = 1, (2, 5) 
The problem I'm having with this is that I have no idea how to solve for x and y...Please help, and thanks in advance!
<pre><font size = 4><b>
Arbitrarily choose x = -5 and 
substitute into

3x + 4y = 1

by replacing x by (-5)

3(-5) + 4y = 1
  -15 + 4y = 1

Add 15 to both sides:

       4y = 1+15
       4y = 16

Divide both sides by 4

       y = 4

So (-5,4) is a point on the line

Arbitrarily choose x = 7 and 
substitute into

3x + 4y = 1

by replacing x by (7)

3(7) + 4y = 1
   21 + 4y = 1

Add -21 to both sides:

       4y = 1-21
       4y = -20

Divide both sides by 4

       y = -5

So (7,-5) is a point on the line

So plot those two points:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
locate(7-.19,-5+.44,o),locate(-5-.19,4+.44,o) 
 )}}} 

Now draw a line through them:

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10,(1-3x)/4 ),
locate(7-.19,-5+.44,o),locate(-5-.19,4+.44,o) 
 )}}} 

Now plot the point (2,5)

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10,(1-3x)/4 ),
locate(7-.19,-5+.44,o),locate(-5-.19,4+.44,o),locate(2-.19,5+.44,o) 
 )}}}

Now construct a perpendicular line from (2,5) to the line

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10,(1-3x)/4 ),
locate(7-.19,-5+.44,o),locate(-5-.19,4+.44,o),locate(2-.19,5+.44,o),
line (2,5,-1,1) 
 )}}}

It appears that this perpendicular touches the line at the point
(-1,1).  Let's stop and make sure that point really is on the line:

Substituting (x,y) = (-1,1)

     3x + 4y = 1
3(-1) + 4(1) = 1
      -3 + 4 = 1
           1 = 1

Yes (-1,1) is on the line.  So all we need to do is to use
the distance formula:
     ___________________
d = <font face = "symbol">Ö</font>(x<sub>2</sub>-x<sub>1</sub>)² + (y<sub>2</sub>-y<sub>1</sub>)²

where (x<sub>1</sub>,y<sub>1</sub>) = (2,5) and (x<sub>2</sub>,y<sub>2</sub>) = (-1,1)
     ________________
d = <font face = "symbol">Ö</font>(-1-2)² + (1-5)²
     _____________
d = <font face = "symbol">Ö</font>(-3)² + (-4)²
     ______
d = <font face = "symbol">Ö</font>9 + 16
     __
d = <font face = "symbol">Ö</font>25

d = 5

So the distance is 5 units.

Edwin</pre>