Learn/Algebra/Linear Functions

Algebra • Topic 12

Linear Functions

Linear functions appear simple ( $y = m x + b$ ), but in Olympiad problems, they often involve optimization, integer solutions (Diophantine equations), or systems with parameters.

Forms of Linear Equations

1. Slope-Intercept Form

y = m x + b

$m$ : Slope (Rate of change).
$b$ : y-intercept.

2. Point-Slope Form

y - y_{1} = m (x - x_{1})

Useful for finding the equation of a line passing through

(x_{1}, y_{1})

with slope

m

3. Standard Form

A x + B y = C

Useful for integer problems (Diophantine equations).
Slope is $- A / B$ .

Properties

1. Parallel and Perpendicular

Parallel Lines: $m_{1} = m_{2}$ .
Perpendicular Lines: $m_{1} m_{2} = - 1$ (Negative reciprocals).

2. Distance from Point to Line

The perpendicular distance

d

from a point

(x_{0}, y_{0})

to the line

A x + B y + C = 0

Theorem (Distance Formula)

d = \frac{∣ A x _{0} + B y _{0} + C ∣}{A ^{2} + B ^{2}}

Linear Diophantine Equations

Equations of the form

a x + b y = c

where

x, y

must be integers.

Theorem (Bézout's Identity)

The equation

a x + b y = c

has integer solutions if and only if

g cd (a, b)

divides

c

If $(x_{0}, y_{0})$ is one solution, the general solution is:

x = x_{0} + \frac{b}{d} t, y = y_{0} - \frac{a}{d} t

where

d = g cd (a, b)

and

t \in Z

Practice Problems

Exercise (Problem 1)

Find the distance from the point

(2, 3)

to the line

3 x - 4 y + 1 = 0

Exercise (Problem 2)

Find all integer pairs

(x, y)

that satisfy

12 x + 51 y = 9

Exercise (Problem 3)

A line passes through

(1, 2)

and creates a triangle of area 4 with the positive x and y axes. Find the equation of the line.