Question

Is the equation -x=1-y and 2+y=x+1 coincident, intersecting, or parallel

Answer 1

$\bf x=1-y\implies x+y=1\implies y=-x+1\\\\ -------------------------------\\\\ 2+y=x+1\implies y=x+1-2\implies y=x-1$

notice the equations in slope-intercept form, the first one has a slope of -1, the second one has a slope of 1.

if the slopes are equal, and the constant is different, they lines are parallel.

if the slopes are equal, and the constant is the same the equations are exactly the same thing, and the lines are coincident, on slapped on top of the other.

if the slopes differ, like here, then they have a solution, where they intersect.