1. The General Equation of a straight line
There are 2 main ways of writing the equation of a straight line:
The gradient/intercept form
y = mx + c
m = gradient
c = y-intercept
General form
ax + by + c = 0
a > 0
a, b, and c are * whole numbers
* “reduced” to lowest integers

2. Example 1 Write y = 3x + ½ in general form. y = 3x + ½ 2y = 6x + 1
What is the gradient and y-intercept of the line 8x + 4y – 12 = 0?
Is the line 8x + 4y – 12 = 0 in general form?
8x + 4y – 12 = 0
4y = –8x + 12
y = –2x + 3
m = –2 and c = 3

3. Gradient between two points
The gradient of a line is a measurement of its steepness.
The gradient is the ratio of the rise to the run of the line.
We use the letter m for gradient. x y
(x1, y1)
(x2, y2)
(y2  y1)
(x2  x1)
For any line.
Pick 2 points (x1, y1) and (x2, y2).
Draw a right angled triangle
Then use “change in y” over “change in x”

4. Gradient between two points
The gradient of a line may be small, having a “gentle” slope
The gradient of a line may be large, having a “steep” slope
A line with a positive gradient slopes “upwards” from left to right.
A line with a negative gradient slopes “downwards” from left to right.

5. Example 2
Show the points A(2, 5), B(6, 11) and C(–6, –7) lie on a straight line
Mathematical word for this is “collinear” First calculate the gradient of any 2 points, say AB.
Then calculate the gradient of either AC or BC
What would happen if we had the points the other way around?.
As mAB = mAC A, B and C are collinear.

6. The point gradient formula
The equation of a straight line through a given point with a given gradient can be found using:
y – y1 = m(x – x1)
m = gradient
(x1, y1) is a point on the line.

7. Example 1
What is the equation of the line through the point (2, 7) with a gradient of ¾?
y – y1 = m(x – x1)
y – 7 = ¾(x – 2)
4y – 28 = 3(x – 2)
4y – 28 = 3x – 6

8. Example 2
What is the equation of the line through the point (3, –4) with a gradient of –5?
y – y1 = m(x – x1)
y – (–4) = –5(x – 3)
y + 4 = –5x + 15
y = –5x + 11