Topic 6

Equation of a Line

A linear equation represents all points

(x, y)

that form a straight line on the coordinate plane. There are several ways to write the equation of a line, depending on what information is given.

1. Slope–Intercept Form

y = mx + b

where:

Example:

y = 2x + 3

has slope

2

and

y

-intercept

3

2. Point–Slope Form

y - y_1 = m(x - x_1)

Used when you know the slope

m

and one point

(x_1, y_1)

on the line. This form is especially useful for building an equation quickly from limited data.

Example: The line through

(2, 5)

with slope

3

y - 5 = 3(x - 2)

which can be rewritten as

y = 3x - 1

3. Standard Form

Ax + By = C

where

A

B

, and

C

are integers, and

A

is usually positive. This form is often used to find intercepts easily:

x\text{-intercept: set } y = 0 \quad \text{and} \quad y\text{-intercept: set } x = 0

Example 1: Writing an Equation from a Graph A line passes through

(0, 2)

and has slope

4

y = 4x + 2

Slope = 4,

y

-intercept = 2.

Example 2: Writing an Equation from Two Points Find the equation of the line through

(1, 3)

and

(5, 11)

Step 1: Find the slope.

m = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2

Step 2: Use point–slope form with $(1, 3)$.

y - 3 = 2(x - 1)

Step 3: Simplify to slope–intercept form.

y = 2x + 1

Final Answer:

\boxed{y = 2x + 1}

Key Takeaways