Find the equation of the line 𝓵 if you know

Question

a). 𝓵 passes across the intersection point of the lines s1 and s2 determined by the equations

x = -5 + x + y and -x + 3y + 1 = -2x + 3y, respectively.

b). 𝓵 is parallel to the line 𝓻 determined by the equation:

-y = 3 - 2x - y - 4(x + 6)

Answer 1

If you want to find an equation of a line, the easiest manner to do that is to find the slope and the y-intercept. We could also use point-slope or two-points. In this case, it might already be defined for us (hint).

Given the following information:

a). 𝓵 passes across the intersection point of the lines s1 and s2 determined by the equations

x = -5 + x + y and -x + 3y + 1 = -2x + 3y, respectively.

b). 𝓵 is parallel to the line 𝓻 determined by the equation:

-y = 3 - 2x - y - 4(x + 6)

The first step, I would suggest is to put all of the equations into a common form for lines.

(a)

x=-5+x+y (subtract an x from both sides) 0=-5+y so y=5

-x + 3y + 1 = -2x + 3y (subtract a 3y from both sides) -x+1=-2x so x=-1

You end up with a horizontal line y=5 and a vertical line x=-1

They clearly intersect at (-1,5)

(b)

-y = 3 - 2x - y - 4(x + 6)

-y = 3 - 2x - y - 4x + -24 (CLT and add y to both sides)

0=-6x-21 so 6x=-21 and x=-3.5

This is also a vertical line at x=-3.5

Therefore, the line you are looking for is the vertical line (necessary to be parallel to x=-3.5) that passes through the intersection point.