Question

Comparing Both Forms
The equation of a line given two points needs to be found. Samuel claims that slope-intercept form will generate the equation and Helena claims that point-slope form will find the equation. Who is correct? Explain your reason by describing both forms.

Answer 1

Both are correct — either form can be used to get the equation of the line through two distinct points (unless the line is vertical, see below). They are just two different algebraic ways to write the same line.

Explain the two forms and how to use them:

- Slope-intercept form: y = mx + b
- m is the slope, b is the y-intercept.
- Given two points (x1,y1) and (x2,y2), first compute the slope m = (y2 − y1)/(x2 − x1) (if x1 ≠ x2). Then plug one point into y = mx + b and solve for b. You then have the line y = mx + b.

- Point-slope form: y − y1 = m(x − x1)
- m is the slope and (x1,y1) is any point on the line.
- Given the two points, compute the slope m = (y2 − y1)/(x2 − x1) and then substitute m and one of the points directly into the point-slope formula. You can leave it in point-slope form or rearrange to slope-intercept form.

Example: points (1,2) and (3,8).
- Slope m = (8−2)/(3−1) = 3.
- Point-slope: y − 2 = 3(x − 1) → y = 3x − 1.
- Slope-intercept: y = 3x + b; plug (1,2): 2 = 3(1) + b → b = −1 → y = 3x − 1.

Special case: if x1 = x2 (vertical line), the slope is undefined and neither slope-intercept nor point-slope (in the usual form) applies; the line is x = x1.

When to prefer which:
- Point-slope is quickest when you have a point and the slope (or two points from which you just computed the slope).
- Slope-intercept is convenient when you want the y-intercept explicitly.

So Samuel and Helena are both right.