Question 128034
For the first problem, remember that lines are parallel if and only if their slopes are equal.  So the first thing to do is find the slope of the given line by putting the equation into slope-intercept form ({{{y=mx+b}}}, i.e. solve the equation for y.

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


Add -3x to both sides:
{{{-y=-3x+4}}}


Multiply by -1:
{{{y=3x-4}}}


Now we know that the slope of the given line is 3 because that is the coefficient on the x term.


If you know a point on a line and the slope of the line, you can use the point-slope form of the line to derive the equation.


{{{y-y[1]=m(x-x[1])}}}, where {{{m}}} is the slope, {{{x[1]}}} is the x-coordinate of the given point, and {{{y[1]}}} is the y-coordinate of the given point.


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


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


{{{y=3x+6}}} is the slope-intercept form of the required equation, or {{{3x-y=-6}}} is the standard form {{{Ax+By=C}}}.


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For the second problem, you are given two points, one of them directly, and the other indirectly by knowing the y-intercept.  The given equation is {{{x+3y=-9}}} and this needs to be put into slope-intercept form by solving for y:


{{{x+3y=-9}}}


{{{3y=-x-9}}}


{{{y=(-x/3)-3}}}.


Now we know that the y-intercept of the given line is -3.  The y-intercept is the point where the line intersects the y-axis.  We know that the x-coordinate of all points on the y-axis is 0, so the ordered pair that we need is (0,-3).


Next we need the two-point form of the line to derive the desired equation:
{{{y-y[1]=((y[1]-y[2])/(x[1]-x[2]))(x-x[1])}}} and the two points: (0,-3) and (-4,-5)

Substituting the coordinate values:
{{{y-(-3)=((-3-(-5))/(0-(-4)))(x-0)}}}


Now do the arithmetic:
{{{y+3=(2/4)(x)}}}


{{{4y+12=2x}}}


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


{{{-x+2y=-6}}}, is the answer you were looking for, though I would have written:


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