Equation of a Line
1. Core Idea
A line's equation is a rule that every point on the line satisfies and no other point does. To write one you need exactly two pieces of information:
- a slope and a point, or
- two points (from which you compute the slope).
2. Must-Know Rules
| Form | Equation | Use when |
|---|---|---|
| Slope-intercept | y = mx + b | you know the slope and the y-intercept |
| Point-slope | y - y1 = m(x - x1) | you know the slope and any point |
| Standard | Ax + By = C | answer choices are in this form |
| Horizontal | y = c | |
| Vertical | x = c |
The workflow for two points:
- Compute the slope m = (y2-y1)/(x2-x1).
- Plug m and either point into point-slope form.
- Rearrange to whatever form the answer choices use.
Testing whether a point lies on a line: substitute its coordinates. If both sides balance, it is on the line.
Finding where two lines intersect: solve the two equations simultaneously. The solution (x, y) is the intersection point.
3. Worked Examples
Example 1 — From two points
Find the equation of the line through (2, 5) and (6, 13).
Slope = (13 - 5)/(6 - 2) = 8/4 = 2.
Point-slope with (2, 5): y - 5 = 2(x - 2) -> y - 5 = 2x - 4
y = 2x + 1.
Check with the second point: 2(6) + 1 = 13. Correct.
Example 2 — Point on a line
Does the point (3, -4) lie on the line 2x - 3y = 18?
2(3) - 3(-4) = 6 + 12 = 18. Yes, it balances.
The point lies on the line.
Example 3 — Intersection
Where do y = 3x - 4 and y = -x + 8 intersect?
Set them equal: 3x - 4 = -x + 8 -> 4x = 12 -> x = 3.
Then y = 3(3) - 4 = 5.
Intersection at (3, 5).
4. GRE Traps
- Sign error in point-slope. y - y1 = m(x - x1). With a point (2, -5), the left side is y + 5.
- Using the point as the intercept. The y-intercept is only the point where x = 0.
- Slope computed with the coordinates in inconsistent order.
- Forgetting to distribute when expanding m(x - x1).
- Answer form mismatch. If the choices are in Ax + By = C form, convert before comparing.
- Assuming a line through the origin. Only if b = 0.
- Vertical lines. They cannot be written as y = mx + b at all; the equation is x = c.
5. Speed Tricks
- Use point-slope form as the default. It works from any point and never requires you to find b separately.
- To check an answer choice quickly, substitute a known point rather than deriving the equation.
- For intersection questions, substitute rather than eliminate when one equation is already solved for y.
- The y-intercept can be read directly if the equation is in y = mx + b form — convert first.
- Two points with the same y give a horizontal line (y = that value). Same x gives a vertical line.
6. Self-Check
Q1. Find the equation of the line with slope -3 passing through (4, 2).
Q2. Find the equation of the line through (0, -5) and (5, 5).
Q3. Where does y = 2x + 1 cross the x-axis?
Answers
A1. y - 2 = -3(x - 4) -> y = -3x + 14.
A2. Slope = 10/5 = 2; y-intercept is -5, so y = 2x - 5.
A3. Set y = 0: 0 = 2x + 1 -> x = -1/2, so (-1/2, 0).
7. One-Line Summary for the Board
Slope plus one point gives the line. Use y - y1 = m(x - x1) and rearrange to whatever form is asked.