Question 954266
f the line passing through the points (a, 2) and (6,7) 

{{{f(x)=mx+b}}} if (6,7), then
{{{7=m*6+b}}}

{{{7=6m+b}}}....eq.1

{{{f(x)=mx+b}}} if (a,2), then

{{{2=m*a+b}}}

{{{2=m*a+b}}}....eq.2

subtract eq.2 from eq 1

{{{7-2=6m-am+b-b}}}....eq.1a

{{{5=(6-a)m}}}
{{{5/(6-a)=m}}}...........eq.a

the line that is parallel to the line passing through the points (4, 8) and (a +3, 2) ,
parallel lines have same slope
{{{f(x)=mx+b}}} if (4,8), then

{{{8=4m+b}}}.......(3)

and (a +3, 2)

{{{2=(a+3)m+b}}}....(4)
subtract eq.4 from eq 3

{{{8-2=4m+b-(a+3)m-b}}}

{{{6=4m-(a+3)m}}}

{{{6=(4-(a+3))m}}}

{{{6=(4-a-3)m}}}

{{{6=(1-a)m}}}

{{{6/(1-a)=m}}}.................eq.a1

from {{{5/(6-a)=m}}}...........eq.a and {{{6/(1-a)=m}}}.................eq.a1 we have:

{{{6/(1-a)=5/(6-a)}}}................solve for {{{a}}}

{{{6(6-a)=5(1-a)}}}
{{{36-6a=5-5a}}}
{{{36-5=6a-5a}}}
{{{31=a}}}

then {{{5/(6-a)=m}}}=>{{{5/(6-31)=m}}}=>{{{5/(-25)=m}}}=>{{{-1/5=m}}}

now we need to find {{{b}}}

 the points (a, 2) and (6,7) are  (31, 2) and (6,7)

{{{f(x)=(-1/5)x+b}}}
{{{2=(-1/5)31+b}}}
{{{2=(-31/5)+b}}}
{{{2+31/5=b}}}
{{{b=41/5}}}


 parallel line {{{g(x)}}}: the points (4, 8) and (a +3, 2) are  (4, 8) and (34, 2)  
{{{f(x)=(-1/5)x+b}}}
{{{8=(-1/5)4+b}}}
{{{8=(-4/5)+b}}}
{{{8+4/5=b}}}
{{{b=44/5}}}

so, your lines are:

{{{f(x)=(-1/5)x+41/5}}} and parallel line {{{g(x)=(-1/5)x+44/5}}}


{{{drawing( 600, 600, -15, 45, -15, 15,
circle(4,8,.25),circle(6,7,.25),circle(31,2,.25),circle(34,2,.25),
locate(4,8,p(4,8)),locate(6,7,p(6,7)),locate(31,2,p(31,2)),locate(34,2,p(34,2)),
 graph( 600, 600, -15, 45, -15, 15, (-1/5)x+41/5,(-1/5)x+44/5)) }}}