Question 373624
Can some please help me !!
(-2,9);3x=7y+6
<pre>


The other tutor made a mistake and used b for m.

Let's draw the given line by getting some points:

 x |  y
-5 | -3
 2 |  0
 9 |  3

{{{drawing(400,400,-6,14,-4,16, graph(400,400,-6,14,-4,16),
locate(-5,-3,"(-5,-3)"), locate(2,0.9,"(2,0)"), locate(9,3,"(9,3)"),
circle(-5,-3,.1), circle(2,0,.1), circle(9,3,.1) )}}}

and then draw a line through them like this one in green:

{{{drawing(400,400,-6,14,-4,16, graph(400,400,-6,14,-4,16),
locate(-5,-3,"(-5,-3)"), locate(2,0.9,"(2,0)"), locate(9,3,"(9,3)"),
circle(-5,-3,.1), circle(2,0,.1), circle(9,3,.1),
green(line(-12,-6,16,6))   )}}}

Now let's plot the point (-2,9) that we want to get the equation of a line
which when graphed will pass through:

{{{drawing(400,400,-6,14,-4,16, graph(400,400,-6,14,-4,16),
locate(-5,-3,"(-5,-3)"), locate(2,0.9,"(2,0)"), locate(9,3,"(9,3)"),
circle(-5,-3,.1), circle(2,0,.1), circle(9,3,.1),
green(line(-12,-6,16,6)), circle(-2,9,.1), locate(-4.5,9,"(-2,9)") 


   )}}}
  
First get the given equation

3x = 7y + 6

into the form y = mx + b

3x = 7y + 6

Write the left side on the right and the right side on the left,
because we want the y to end up on the left side.

7y + 6 = 3x

Add -6 to both sides:

7y + 6 = 3x
   - 6      - 6
---------------
7y     = 3x - 6

Now divide every coefficient and the number 6 by 7

{{{7/7}}}y = {{{3/7}}}x - {{{6/7}}}

y = {{{3/7}}}x - {{{6/7}}}

That is now in the form y = mx + b

and we can look at the coefficient of x and see that
the slope m is {{{3/7}}}

All parallel lines have the same slope, so if we want to
get a line parallel to that one we will use the same slope

So the line we want the equation of will have the same 
slope {{{3/7}}} but it will have a different value for b.

So we write

y = {{{3/7}}}x + b

Now we will substitute the x-value of the point (-2,9) which
is -2 for x, and we will substitute the y-value of the point
(-2,9), which is 9, for y in that equation:

9 = {{{3/7}}}(-2) + b

Now we will multiply through by 7 to get rid of the fraction:

7*9 = 7*{{{3/7}}}(-2) + 7*b

 63 = 3(-2) + 7b
 63 = -6 + 7b
 
Add +6 to both sides

 63 = -6 + 7b
 +6   +6
 ------------
 69 =      7b

Now we will divide both sides by 7

{{{69/7}}} = {{{7b/7}}}

{{{69/7}}} = b

Finally we will substitute {{{69/7}}} for b in y = {{{3/7}}}x + b
and the desired equation is

y = {{{3/7}}}x + {{{69/7}}}

Now let's check by getting a couple points on the desired line
  
Let's draw the final line by getting some points:

 x |  y
 5 | 12
12 | 15
-2 |  9

{{{drawing(400,400,-6,14,-4,16, graph(400,400,-6,14,-4,16),
locate(-5,-3,"(-5,-3)"), locate(2,0.9,"(2,0)"), locate(9,3,"(9,3)"),
circle(-5,-3,.1), circle(2,0,.1), circle(9,3,.1),
green(line(-12,-6,16,6)), circle(-2,9,.1), locate(-4.5,9,"(-2,9)"),

locate(5,12,"(5,12)"), circle(5,12,.1), 
locate(9,15,"(12,15)"), circle(12,15,.1)

   )}}}

Now let's draw a line through those points and see if it looks
like the line (in red) is parallel to the green one:

{{{drawing(400,400,-6,14,-4,16, graph(400,400,-6,14,-4,16),
locate(-5,-3,"(-5,-3)"), locate(2,0.9,"(2,0)"), locate(9,3,"(9,3)"),
circle(-5,-3,.1), circle(2,0,.1), circle(9,3,.1),
green(line(-12,-6,16,6)), circle(-2,9,.1), locate(-4.5,9,"(-2,9)"),

locate(5,12,"(5,12)"), circle(5,12,.1), 
locate(9,15,"(12,15)"), circle(12,15,.1),
red(line(-16,3,19,18)) 

   )}}}

The red line looks parallel to the green line, so we are right.


Edwin</pre>