Question 129083
Step 1:  Solve the given equation for y, putting it into slope-intercept form ({{{y=mx+b}}}).


{{{7x+y=-3}}}
{{{y=-7x-3}}}


Step 2:  By inspection of the coefficient on x of the slope-intercept form, determine the slope of the given line:  {{{m=-7}}}.


Step 3:  Two lines are parallel if and only if their slopes are equal, i.e. {{{m[1]=m[2]}}}.  Therefore, the desired line must also have a slope of {{{m=-7}}}


Step 4:  Use the point-slope form of the line ({{{y-y[1]=m(x-x[1])}}}, the x- and y-coordinates of the given point, and the slope number from Step 3 to derive an equation for the desired line.


The given point is P(1,4), so {{{x[1]=1}}} and {{{y[1]=4}}}.


{{{y-4=-7(x-1)}}}


Step 5:  Put this new equation into standard form.  Standard form is {{{Ax+By=C}}}.


Distribute the -7 and remove parentheses:
{{{y-4=-7x+7}}}


Add 4 to both sides:
{{{y=-7x+11}}}


Add 7x to both sides:
{{{7x+y=11}}}


Done.


The red line is the given line, and the green one is the derived one.  At least they look like they are parallel.


{{{drawing(600,600,-15,15,-15,15,
grid(1),
graph(600,600,-15,15,-15,15,-7x-3,-7x+11)
)}}}