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A linear equation is an equation for a straight line

These are all linear equations:

yes  y = 2x + 1

yes  5x = 6 + 3y

yes  y/2 = 3 − x

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:

xy = 2x + 1

-1y = 2 × (-1) + 1 = -1

0y = 2 × 0 + 1 = 1

1y = 2 × 1 + 1 = 3

2y = 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").

Examples: These are linear equations:

yes  y = 3x − 6

yes  y − 2 = 3(x + 1)

yes  y + 2x − 2 = 0

yes  5x = 6

yes  y/2 = 3

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:

not  y2 − 2 = 0

not  3√x − y = 6

not  x3/2 = 16

Slope-Intercept Form

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

y=mx+b graph

Equation of a Straight Line y=mx+b

Slope (or Gradient)Y Intercept

Example: y = 2x + 1

Slope: m = 2

Intercept: b = 1

Animation    

Play With It !

You can see the effect of different values of m and b at Explore the Straight Line Graph

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:

InOut

00

55

−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

Source: https://www.mathsisfun.com/algebra/linear-equations.html

Step-by-step explanation:

hee hmu ;) <333333333