GRADE 9 Mathematics ALGEBRA – EQUATION OF STRAIGHT LINES Notes
Mathematics — ALGEBRA
Subtopic: Equation of Straight Lines (Age ~14, Kenya)
In algebra, a straight line on the coordinate plane can be described by an equation. These notes explain common forms, how to find the equation from points or slope, and how to sketch a line.
Coordinate plane quick recap:
- Horizontal axis = x-axis, vertical axis = y-axis.
- Each point has coordinates written as (x, y).
1. Slope (Gradient) of a line
The slope m tells how steep a line is. For two points (x1, y1) and (x2, y2):
m = (y2 − y1) / (x2 − x1)
Example: points (1, 2) and (4, 5)
m = (5 − 2) / (4 − 1) = 3 / 3 = 1
2. Common forms of a straight line
a) Slope-intercept form
y = mx + c
- m is the slope (gradient).
- c is the y-intercept (point where the line crosses the y-axis).
y = mx + c
- m is the slope (gradient).
- c is the y-intercept (point where the line crosses the y-axis).
Example: y = 2x + 1 → slope m = 2, y-intercept c = 1 (point (0,1))
b) Point-slope form
y − y1 = m(x − x1)
- Use when you know a point (x1, y1) and the slope m.
Example: slope 3 through point (2, −1): y + 1 = 3(x − 2).
y − y1 = m(x − x1)
- Use when you know a point (x1, y1) and the slope m.
Example: slope 3 through point (2, −1): y + 1 = 3(x − 2).
c) Two-point form
If you know two points (x1, y1) and (x2, y2):
(y − y1) / (x − x1) = (y2 − y1) / (x2 − x1)
Rearranged it becomes point-slope with slope computed from the two points.
If you know two points (x1, y1) and (x2, y2):
(y − y1) / (x − x1) = (y2 − y1) / (x2 − x1)
Rearranged it becomes point-slope with slope computed from the two points.
d) General form
ax + by + c = 0 (a, b, c are constants).
You can rearrange between general form and slope-intercept: y = mx + c.
ax + by + c = 0 (a, b, c are constants).
You can rearrange between general form and slope-intercept: y = mx + c.
3. Finding the equation from two points — worked example
Find the equation of the line through A(1, 2) and B(4, 5).
- Find slope: m = (5 − 2)/(4 − 1) = 3/3 = 1.
- Use point-slope with point A(1,2): y − 2 = 1(x − 1).
- Simplify: y − 2 = x − 1 ⇒ y = x + 1.
4. Intercepts
- y-intercept: set x = 0, solve for y (gives point (0, c) in y = mx + c).
- x-intercept: set y = 0, solve for x (solve 0 = mx + c ⇒ x = −c/m if m ≠ 0).
5. Parallel and perpendicular lines
- Parallel lines have the same slope. Example: y = 2x + 1 and y = 2x − 3 are parallel.
- Perpendicular lines have slopes that multiply to −1. If one slope is m, the perpendicular slope is −1/m. Example: slope 2 → perpendicular slope −1/2.
6. Simple graph illustration
Below is a small sketch of the line y = 0.5x + 1 (for visual understanding):
7. Short practice (try these)
- Find the slope of the line through (2, 3) and (5, 9).
- Write the equation of the line with slope −2 that passes through (1, 4).
- Find the equation of the line through (0, −2) and (3, 1).
- Are the lines y = 3x + 5 and y = 3x − 2 parallel, perpendicular or neither?
Answers (check after trying):
- Slope = (9 − 3)/(5 − 2) = 6/3 = 2.
- Use y − 4 = −2(x − 1) ⇒ y − 4 = −2x + 2 ⇒ y = −2x + 6.
- Slope = (1 − (−2)) / (3 − 0) = 3/3 = 1 ⇒ y = x − 2.
- Parallel (same slope 3).
8. Tips for exams (KCSE/School tests)
- Always calculate slope carefully; swap subtraction order consistently (y2−y1)/(x2−x1).
- To graph quickly from y = mx + c: start at (0, c) then use rise/run = m.
- Check special cases: vertical lines (x = k) have no slope (undefined); horizontal lines (y = k) have slope 0.
If you want, I can make more worked examples, printable exercise sheets, or short quizzes on equations of straight lines tailored to the Kenyan syllabus—tell me which you prefer.