Question

Finding the Equation of a Line

You have discovered another stranded alien.

What is the equation of the line that provides the correct path to rescue this alien?

Answer 1

I don't have the coordinates or the picture of the alien path, so I can't give a single numeric equation yet. Please tell me one of the following (or upload the image):

- two points on the path (x1,y1) and (x2,y2), or
- one point and the slope m, or
- the x- and y-intercepts, or
- the graph/image.

How to get the equation once you have that:

1. From two points (x1,y1) and (x2,y2)
- Compute slope: m = (y2 − y1) / (x2 − x1) (if x2 = x1 the line is vertical: x = x1).
- Use point–slope form: y − y1 = m(x − x1).
- Convert to slope–intercept form y = mx + b by solving for b = y1 − m x1, or to standard form Ax + By + C = 0.

2. From one point (x1,y1) and slope m
- Use point–slope: y − y1 = m(x − x1), then rearrange if desired.

3. From intercepts x-intercept a and y-intercept b
- Equation: x/a + y/b = 1 (unless an intercept is 0; handle vertical/horizontal separately).

Examples
- Points (1,2) and (4,5): m = (5−2)/(4−1) = 1, so y − 2 = 1(x − 1) → y = x + 1.
- Point (3,−2) with m = 2: y + 2 = 2(x − 3) → y = 2x − 8.

Send the coordinates or the image and I’ll compute the exact equation for the rescue path.