Equations Of Lines

Before viewing this page, it would be helpful to learn about the Gradient (or Slope).

Linear Equations as y = mx + c

The most commonly used form of a linear equation is y = mx + c

where
m is the gradient
c is the y-intercept.


Example One - y = mx + c

Find the gradient m and the y-intercept c of the following equations:
(a) y = 3x + 4
(b) y = 5x – 2
(c) y = x

Answer:
(a) gradient = 3, y-intercept = 4
(b) gradient = 5, y-intercept = –2
(c) gradient = 1, y-intercept = 0


Linear Equations in Standard Form

The standard form of linear equation is ax + by = c

In standard form, note that c in this form is not the y-intercept.


Example Two - Standard Form

Write the equation y = 3x + 5 in standard form.

Answer:
y = 3x + 5
–3x + y = 5

Traditionally in standard form, the coefficient of the x is always a positive number. To achieve this, multiply the whole equation by –1.
3x – y = –5


Example Three - Find Linear Equations

Use the formula: y – y1 = m (x – x1)

The gradient of a line is 2. The line passes through the point (4,5). What is the equation of this line?

Answer:
(x1, y1) = (4,5)
m = 2

y – y1 = m ( x – x1 )
y – 5 = 2 ( x – 4 )
y – 5 = 2x – 8
y = 2x – 8 + 5
y = 2x – 3


Example Four - Find Linear Equations

A line passes through the points (4,5) and (7, 16). What is the equation of this line?
(Hint: Find the gradient first.)

Answer:
(x1, y1) = (4,5)
(x2, y2) = (7,16)

m = y2 – y1 = 16 – 5 = 9 = 3
x2 – x17 – 43

y – y1 = m ( x – x1 )
y – 5 = 3 ( x – 4 )
y – 5 = 3x – 12
y = 3x – 12 + 5
y = 3x – 7