Question 208010
determine whether each pair of equations represents parallel lines:
y=-3x+1, 6x+2y=8

<pre><font size = 4 color = "indigo"><b>
There are several ways to do this. Pick whichever way
your teacher told you to use.

Method 1:  Put them both in slope-intercept form. If 
their slopes are equal, but they have different 
y-intercepts, then they are parallel.

Method 2:  Choose any two values for x.  Substitute 
           them both in each equation to find the 
           y-values corresponding to them.  If the 
           corresponding y-values have the same non-zero
           difference, then the lines are parallel.

Method 3:  Put them both in general form. Then 

{{{system(A[1]x+B[1]y=C[1],A[2]x+B[2]=C[2])}}}

if {{{A[1]/A[2]=B[1]/B[2]}}} then they are parallel.   

----------------------

Method 1:  

{{{system(y=-3x+1, 6x+2y=8)}}}

The first one is already in the slope intercept form
{{{y=mx+b}}}.  Get the second one in slope-intercept 
form, by solving for y

{{{6x+2y=8}}}

Add {{{-6x}}} to both sides:

{{{2y=-6x+8}}}

Divide every term by {{{2}}}:

{{{(2/2)y =((-6)/2)x+(8/2)}}}

{{{y=-3x+4}}}

So the slope is the coefficient of x, which is -3
So the value of the slope {{{m}}} is the same in 
both equations, but their y-intercepts are different.
Therefore the lines those equations
represent are parallel.

----------------------------------

Method 2:

{{{system(y=-3x+1, 6x+2y=8)}}}

Substitite x=0 in each and solve for y:

In the first one:

{{{y=-3x+1}}}  
{{{y=-3(0)+1}}}
{{{y=1}}}

In the second one:

{{{6x+2y=8}}}
{{{6(0)+2y=8}}}
{{{0+2y=8}}}
{{{2y=8}}}
{{{y=4}}}

Subtract them, second y - first y = {{{4-1=3}}}

Substitite x=1 in each and solve for y:

In the first one:

{{{y=-3x+1}}}  
{{{y=-3(1)+1}}}
{{{y=-3+1}}}
{{{y=-2}}}

In the second one:

{{{6x+2y=8}}}
{{{6(1)+2y=8}}}
{{{6+2y=8}}}
{{{2y=2}}}
{{{y=1}}}

Subtract them, second y - first y = {{{1-(-2)=1+2=3}}}

We get the same difference 3, so the lines represented
by those equations are parallel.

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Method 3:

{{{system(y=-3x+1, 6x+2y=8)}}}

Get the forst one in general form:

{{{y=-3x+1}}}
{{{3x+y=1}}}

So the system is now:

{{{system(3x+y=1, 6x+2y=8)}}}

See if the coefficients of x and y are in
the same proportion:

Is {{{3/6=1/2}}} true?

Yes it is, therefore the lines those equations
represent are parallel.

Edwin</pre>