Question

A line passes through the point (–3, –4) and its y-intercept is (0, –9). What is the equation of the line that is parallel to the first line and passes through the point (3, –7)?

Answer 1

Let's begin by finding the equation that has those two first points on it.
It passes through (-3,-4) and (0.9), the y-intercept is also a point, so let's consider it that way.
We have to find the slope wich is calculated by the difference in the y axis, divided by the difference in the x axis, written as "m" and defined by the following formula:
\[m=\frac{ \Delta y }{ \Delta x }=\frac{ y_2 - y_1 }{ x_2 - x_1 }\]
so we have the two points given by the letter, and let's apply it, let's call point 1(-3.-4) and point 2(0,9).
In order to calculate the slope we must apply the formula I gave you above:
\[m=\frac{ 0-(-4) }{ 9-(-3) }=\frac{ 4 }{ 12 }=\frac{ 1 }{ 4 }\]
by transitivity:
\[m=\frac{ 1 }{ 4 }\]
Now that we have the slope corresponding to the line, let's apply the formula that allows me to find the line knowing the slope and a point belonging to it.
The formula looks like this:
\[(y-y_p ) = m(x- x_p )\]
Where "m" is the slope and "(xp,yp)" is any known point.
Since we have the formula, and two points, we can take any of those and plug them in, to find the corresponding line equation.
I'll go with point 1 and the slope we just found:
\[(y-9)=(\frac{ 1 }{ 4 })(x-0)\]
Doing the corresponding operations and simplifications:
\[y=\frac{ x }{ 4 }+9\]
I personally like to put it in general form, so I'll do that, but you can keep the equality above wich works just fine:
\[-x+4y-36=0\]
Would be tje line that goes through (-3,-4) and has a y- intercept of (0,9).