Verified answer

Parallel lines (assuming 2D) will have the same slope. So it helps a lot to put the equations into standard form (slope-intercept form.) This is with y on the left and everything else on the right so that there is a multiplier of x and an added value. (The standard form is usually written as y=m*x+b, such that you need to figure out m and b.) If you have two line equations and they both have the same value of m, then they are parallel. (The only problem here is that if the two lines are parallel but vertical, then m will be infinitely large... but you can figure that out, too.)

Assuming your two equations in the problem are for two distinct, hypothetical lines to compare, let's put the both into standard form. The first one is 2y+2=x. I will list out the steps I used:

2y+2=x

2y=x-2, by subtracting 2 from both sides

y=(1/2)*x - 1, by dividing 2 into both sides.

The second one is (1/2)x=-1+y. Steps listed:

(1/2)x=-1+y

y-1=(1/2)x, by flipping the sides and reordering the -1+y into y-1

y=(1/2)x+1, by adding 1 to both sides.

Note that the multipler (m) of x in both equations is the same. So the lines are parallel. By noting that the intercept (b) is -1 for the first and 1 for the second, you know they aren't the SAME line. So they don't coincide. But they are parallel. And being parallel they also don't intersect. (Well, unless you are talking about spherical/Riemann geometry.)

If they aren't parallel, then they must intersect at some point. So that's how you tell.... just look at the multiplier of x in the two equations and if they are different then they will intersect. No need to look at the intercept value (b).

You didn't ask how to find that point, so I won't add that answer here.

if a line intersects or coincides there is solution x,y when solving the equations for both the lines.

x,y would give the intersection point

if the line is parallel, there is a solution when x=k or y=k

you find the slope. if the slope is the same, it is going the same direction, and therefore parallel. if it is the opposite of the reciprocal of the other slope, it is perpendicular.

y=5x+8 and y=5x+9 are parallel

y=5x+8 and y=(-1/5)x+6 are perpendicular

its easier when you put them into slope intercept form like this.

y=mx+b where m is the slope and b is the y-intercept.