Question

I need help with coming up with my own equations for intersecting lines, parallel lines, and the same line.

Answer 1

straight lines always follows the same format in Cartesian space - ie the 2 dimensional co-ordinate system, where points in the space are identified as (a,b).

a typical equation: y = mx + x

m is the gradient which the ratio of how much a line rises in the vertical y axis/direction to how much it travels across the x axis/direction.

this means that any of these following lines will never meet because they are parallel:
y = 2x-3
y = 2x+5
y= 2x+100

they each have a gradient of +2 and will therefore never ever meet.

[a line with a negative gradient falls as it travels out on the x axis. eg y = -2 +3]

that's the m in y = mx + c.

the c is the intercept, ie where the live touches the y axis at x = 0.

so in the first example of y = 2x - 3, when x = 0, y = 3. In the second, when x = 0, y = 5. we now know that these line sit one above the other. they have the same gradient and will never ever meet because they are parallel, and one sits 2 units above the other on the y axis.