Linear Equations

A linear equation is an equation for a straight line

These are all linear equations:

Let us look more closely at one example:

Example: y = 2x + 1 is a linear equation:

line on a graph

The graph of y = 2x+1 is a straight line

  • When x increases, y increases twice as fast, so we need 2x
  • When x is 0, y is already 1. So +1 is also needed
  • And so: y = 2x + 1

Here are some example values:

x y = 2x + 1
-1 y = 2 × (-1) + 1 = -1
0 y = 2 × 0 + 1 = 1
1 y = 2 × 1 + 1 = 3
2 y = 2 × 2 + 1 = 5

Check for yourself that those points are part of the line above!

Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

But the variables (like "x" or "y") in Linear Equations do NOT have:

  • Exponents (like the 2 in x2)
  • Square roots, cube roots, etc

Examples: These are NOT linear equations:

Slope-Intercept Form

The most common form is the slope-intercept equation of a straight line:

y=mx+b graph

Example: y = 2x + 1

  • Slope: m = 2
  • Intercept: b = 1

Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

y − y1 = m(x − x1)

Point-Slope Form

Example: y − 3 = (¼)(x − 2)

It is in the form y − y1 = m(x − x1) where:

  • y1 = 3
  • m = ¼
  • x1 = 2

General Form

And there is also the General Form of the equation of a straight line:

Ax + By + C = 0
(A and B cannot both be 0)

Example: 3x + 2y − 4 = 0

It is in the form Ax + By + C = 0 where:

  • A = 3
  • B = 2
  • C = −4

There are other, less common forms as well.

As a Function

Sometimes a linear equation is written as a function, with f(x) instead of y:

y = 2x − 3
f(x) = 2x − 3
These are the same!

And functions are not always written using f(x):

y = 2x − 3
w(u) = 2u − 3
h(z) = 2z − 3
These are also the same!

The Identity Function

There is a special linear function called the "Identity Function":

f(x) = x

And here is its graph:

Identity Function
It makes a 45° (its slope is 1)

It is called "Identity" because what comes out is identical to what goes in:

In Out
0 0
5 5
−2 −2
...etc ...etc

Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

Constant Function

f(x) = C

No matter what value of "x", f(x) is always equal to some constant value.

Using Linear Equations

You may like to read some of the things you can do with lines:

  • Finding the Midpoint of a Line Segment
  • Finding Parallel and Perpendicular Lines
  • Finding the Equation of a Line from 2 Points